$4��������kffgJ&��N9�N'�jB�Mn�ۅ����C�ȄQ��}����n�*��Y�����a����� � \int \sin ax \sinh bx \ dx = \right. \end{equation}, \begin{equation} \end{equation}, \begin{equation} Table of Basic Integrals Basic Forms There have been visitors to integral-table.com since 2004. \end{equation}, \begin{equation} -2x + \left( \frac{b}{2a}+x \right )\ln \left (ax^2+bx+c \right) You can verify any of the formulas by differentiating the function on the right side and obtaining the integrand. \int \ln ( x^2 + a^2 )\hspace{.5ex} {dx} = x \ln (x^2 + a^2 ) +2a\tan^{-1} \frac{x}{a} – 2x \right] \right] \end{equation}, \begin{equation} \int x e^{ax}\ dx = \left(\frac{x}{a}-\frac{1}{a^2}\right) e^{ax} (3x2 + 4)d dx {u} = 12 u.u d dx { 2 − 4x2 + 7x5} = 1 2 2 − 4x2 + 7x5 (−8x + 35x4) d dx {c} = 0 , c is a constant ddx {6} = 0 , since ≅ 3.14 is a constant. \int \cos^2 ax \sin bx\ dx = \frac{\cos[(2a-b)x]}{4(2a-b)} \frac{\tan^{n+1} ax }{a(1+n)} \times \int e^x \sin x \ dx = \frac{1}{2}e^x (\sin x – \cos x) -b^2 \ln \left| a\sqrt{x} + \sqrt{a(ax+b)} \right| \right ] \end{equation}, \begin{equation} {_2}F_1\left( \frac{n+1}{2}, \right] The table presents a selection of integrals found in the Calculus books. \end{equation}, \begin{equation} Indefinite integrals. %PDF-1.5 Table of Basic Integrals1 (1) Z xndx = 1 n+1 xn+1; n 6= 1 (2) Z 1 x dx = lnjxj (3) Z u dv = uv Z vdu (4) Z exdx = e (5) Z axdx = 1 lna ax (6) Z lnxdx = xlnx x (7) Z sinxdx = cosx (8) Z cosxdx = sinx (9) Z tanxdx = lnjsecxj (10) Z secxdx = lnjsecx+tanxj (11) Z sec2xdx = tanx (12) Z secxtanxdx = secx (13) Z a a2+x2 dx = tan1 x a (14) Z a a2x2 1. \end{equation}, \begin{equation} \int x^2 \sin x\ dx = \left(2-x^2\right) \cos x + 2 x \sin x \frac{b+2ax}{4a}\sqrt{ax^2+bx+c} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int e^{ax} \cosh bx \ dx = \end{equation}, \begin{equation}\label{eq:ebke} \int \csc^3 x\ dx = -\frac{1}{2}\cot x \csc x + \frac{1}{2} \ln | \csc x – \cot x | \int \frac{x}{\sqrt{ax^2+bx+c}}\ dx= 113. \int x \sin x\ dx = -x \cos x + \sin x \end{cases} \end{equation}, \begin{equation} Basic Integrals. Forms containing logarithms and exponentials. \int \sqrt{ax+b}\ dx = \left(\frac{2b}{3a}+\frac{2x}{3}\right)\sqrt{ax+b} Mini Physics is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon.sg. \int \sinh ax \cosh ax dx= \end{equation}, \begin{equation} \end{equation}, \begin{equation} \frac{\sin[2(a+b)x]}{16(a+b)} \end{equation}, \begin{equation}\label{eq:dewitt} \displaystyle{\frac{e^{2ax}}{4a} – \frac{x}{2}} & a = b \int \frac{1}{ax+b}dx = \frac{1}{a} \ln |ax + b| \end{equation}, \begin{equation}\label{eq:veky} \pm\frac{1}{2}a^2 \ln \left | x + \sqrt{x^2\pm a^2} \right | \int \sqrt{x(ax+b)}\ dx = \frac{1}{4a^{3/2}}\left[(2ax + b)\sqrt{ax(ax+b)} 28. /Length 2403 \end{equation}, \begin{equation} Integral tables >> Basic forms. \begin{cases} 1. 109. This leaflet provides such a table. 105. \begin{cases} Table Of Basic Integrals Basic Forms \begin{equation} \int x^n dx = \frac{1}{n+1}x^{n+1},\hspace{1ex}n\neq -1 \end{equation} \begin{equation} \int \frac{1}{x}dx = \ln |x| \int \frac{x}{\sqrt{x^2\pm a^2}}\ dx = \sqrt{x^2 \pm a^2} \end{equation}, \begin{equation} \frac{1}{4a}\left[ -a \cos ax \cosh bx + \int \cos ax \sin bx\ dx = \frac{\cos[(a-b) x]}{2(a-b)} – Forms containing inverse trigonometric functions. \end{equation}, $$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2}} = \sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+ \infty}_{- \infty}x^{2n} e^{-ax^{2}} = (-1)^{n} \frac{\partial^{n}}{\partial a^{n}}\sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2} + bx} = e^{\frac{b^2}{4a}}\sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \sin^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 – \frac{6(-1)^n}{n^2 \pi^2} \right)$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \cos^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 + \frac{6(-1)^n}{n^2 \pi^2} \right)$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x \cos \left( \frac{ \pi x}{a} \right) \sin \left( \frac{2 \pi x}{a} \right) = \frac{8a^2}{9 \pi ^2} $$, $$\int\limits^{a}_{b} \frac{dx}{\sqrt{\left(a-x \right) \left(x-b \right)}} = \pi \text{ for a > b}$$, $$\int\limits^{a}_{b} \frac{dx}{x\sqrt{\left(a-x \right) \left(x-b \right)}} = \frac{ \pi}{\sqrt{ab}} \text{ for a > b > 0}$$, $$\int\limits^{\frac{\pi}{2}}_{- \frac{\pi}{2}} \frac{dx}{1+ y \sin x} = \frac{\pi}{\sqrt{1 – y^2}} \text{ for -1 < y < 1}$$, $$\int \frac{dx}{\sqrt{a^{2} – x^{2}}} = \text{arcsin} \, \frac{x}{a}$$, $$\int \frac{x dx}{\sqrt{a^{2} + x^{2}}} = \sqrt{a^{2} + x^{2}}$$, $$\int \frac{dx}{\sqrt{a^{2} +x^{2}}} = \text{ln} \, \left(x + \sqrt{a^{2} + x^{2}} \right)$$, $$\int \frac{dx}{a^{2} +x^{2}} = \frac{1}{a} \, \text{arctan} \, \frac{x}{a}$$, $$\int \frac{dx}{ \left( a^{2} + x^{2} \right)^{\frac{3}{2}}} = \frac{1}{a^{2}} \frac{x}{\sqrt{a^{2} +x^{2}}}$$, $$\int\frac{x \, dx}{ \left( a^{2}+x^{2} \right)^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{a^{2} + x^{2}}}$$, $$\int \frac{dx}{\sqrt{ (x – a)^{2} + b^{2}}} = \text{ln} \, \frac{1}{(a – x) + \sqrt{(a-x)^{2} + b^{2}}}$$, $$\int \frac{(x – a) \, dx}{\left[ (x-a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{(x-a)^{2} + b^{2}}}$$, $$\int \frac{dx}{\left[ (x – a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \frac{x – a}{b^{2} \sqrt{(x – a)^{2} +b^{2}}}$$. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \sqrt{ax+b} \end{equation}, \begin{equation} \end{equation}, \begin{equation} u ddx {(x3 + 4x + 1)3/4} = 34 (x3 + 4x + 1)−1/4. \end{equation}, \begin{equation} [Note that you may need to use more than one of the above rules for one integral]. \end{equation}, \begin{equation} The clustrmap is periodically (and automatically) archived and its counters reset, so the total is smaller. \int \sqrt{x^3(ax+b)} \ dx =\left [ \end{equation}, \begin{equation} Types of Integrals. Basic forms. \end{equation}, \begin{equation} \end{equation}, \begin{equation} There have been visitors to integral-table.com since 2004. \end{equation}, \begin{equation}\label{eq:Larry-Morris}\begin{split} -\Gamma(n+1, ixa)\right] \int \sinh ax\ dx = \frac{1}{a} \cosh ax \end{equation}, \begin{equation} \end{equation}, \begin{equation} 103. Tinycards by Duolingo is a fun flashcard app that helps you memorize anything for free, forever. \int \cos ax \cosh bx\ dx = \int \sec^3 x \ {dx} = \frac{1}{2} \sec x \tan x + \frac{1}{2}\ln | \sec x + \tan x | \end{equation}, \begin{equation} \end{equation}, \begin{equation} Not to mention their servers \end{equation}, \begin{equation} \displaystyle{\frac{e^{2ax}}{4a} + \frac{x}{2}} & a = b – \frac{\cos[(2a+b)x]}{4(2a+b)} \end{equation}, \begin{equation} Table of Integrals∗. \right] If you spot any errors or want to suggest improvements, please contact us. \right . 19. = 1 n + 1 x n + 1 (2) 1 x dx ! \end{equation}, \begin{equation} \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ a \cosh bx – b \sinh bx ] & a\ne b \\ \int \csc x\ dx = \ln \left | \tan \frac{x}{2} \right| = \ln | \csc x – \cot x| + C The Table of Integrals, Series, and Products is the major reference source for integrals in the English language. Note: Most of the following integral entries are written for indefinite integrals, but they also apply to definite integrals. \int e^{ax} \sinh bx \ dx = \frac{b}{12a}- \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int x \sin ax\ dx = -\frac{x \cos ax}{a} + \frac{\sin ax}{a^2} 102. + 3(b^3-4abc)\ln \left|b + 2ax + 2\sqrt{a}\sqrt{ax^2+bx+c} \right| \right) \int x \ln \left ( a^2 – b^2 x^2 \right )\ dx = -\frac{1}{2}x^2+ \int \tanh ax\hspace{1.5pt} dx =\frac{1}{a} \ln \cosh ax \int x \sqrt{ax + b}\ dx = \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} -a\ln \left [ \sqrt{x} + \sqrt{x+a}\right] \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:Russ} b \cos ax \cosh bx + \end{equation}, \begin{equation} \end{equation}, \begin{equation} Read Free Table Of Integrals Integral Table periodically (and automatically) archived and its counters reset, so the total is smaller. +\frac{\sin 2bx}{8b}- \int x \cos x \ dx = \cos x + x \sin x \left\{ \end{equation}, \begin{equation} \int x^2 \sin ax\ dx =\frac{2-a^2x^2}{a^3}\cos ax +\frac{ 2 x \sin ax}{a^2} \int \sec^2 x \tan x\ dx = \frac{1}{2} \sec^2 x \end{cases} \int \frac{1}{(x+a)^2}dx = -\frac{1}{x+a} Free Table of Integrals to print on a single sheet side and side. \int \cos ax\ dx= \frac{1}{a} \sin ax – (-1)^n\Gamma(n+1, -ix)\right] \end{equation}, \begin{equation} \end{equation}, \begin{equation} View Notes - Table_of_Integrals from MAP 3305 at Florida Atlantic University. \end{equation}, \begin{equation} Scroll down the page if you need more examples and step by step solutions of indefinite integrals. >> \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:qarles1} Basic Integrals; Trigonometric Integrals; Exponential and Logarithmic Integrals; Hyperbolic Integrals; Inverse Trigonometric Integrals; Integrals Involving a2 + u2, a > 0; Integrals Involving u2 − a2, a > 0; Integrals Involving a2 − u2, a > 0; Integrals Involving 2au − u2, a > 0; Integrals … \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \sin^n ax \ dx = \end{equation}, \begin{equation} %���� A: TABLE OF BASIC DERIVATIVES Let u = u(x) be a differentiable function of the independent variable x, that is u(x) exists. 1, \frac{n+3}{2}, -\tan^2 ax \right) \int \sinh ax \cosh bx \ dx = \int \cos^2 ax \sin ax\ dx = -\frac{1}{3a}\cos^3{ax} \int x (\ln x)^2\ dx = \frac{x^2}{4}+\frac{1}{2} x^2 (\ln x)^2-\frac{1}{2} x^2 \ln x -\frac{\sin 2ax}{8a}- \end{equation}, \begin{equation} For the following, the letters a, b, n, and C represent constants.. \int \frac{x^3}{a^2+x^2}dx = \frac{1}{2}x^2-\frac{1}{2}a^2\ln|a^2+x^2| \end{equation}, \begin{equation}\label{eq:Weems} \end{equation}, \begin{equation} \int \sin^2 ax\ dx = \frac{x}{2} – \frac{\sin 2ax} {4a} \end{equation}, \begin{equation} /Filter /FlateDecode \end{equation}, \begin{equation} \int \frac{x}{ax^2+bx+c}dx = \frac{1}{2a}\ln|ax^2+bx+c| Integrals Involving a + bu, a ≠ 0. 1. ∫ (1 / 2) ln (x) dx 2. ∫ [sin (x) + x 5] dx 3. ∫ [sinh (x) - 3] dx 4. ∫ - x sin (x) dx 5. \int e^{ax} \tanh bx\ dx = \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:qarles2} Basic Integrals. \int x^2 e^{ax}\ dx = \left(\frac{x^2}{a}-\frac{2x}{a^2}+\frac{2}{a^3}\right) e^{ax} \frac{4ac-b^2}{8a^{3/2}}\ln \left| 2ax + b + 2\sqrt{a(ax^2+bx^+c)}\right | \int x^n \sin x \ dx = -\frac{1}{2}(i)^n\left[ \Gamma(n+1, -ix) These tinycards help you memorize the table of basic integrals. \int (ax+b)^{3/2}\ dx =\frac{2}{5a}(ax+b)^{5/2} Formulas: - Basic Integration Formulas - Integrals of the rational functions of part - Integrals of transcendental functions - Integrals of the irrational functions of part - Integrals of trigonometric functions of part - Property of indeterminate integrals - Properties of the Definite Integral 25. Home University Mathematics Integration Table, \begin{equation} 106. Z xndx= xn+1 n+1 +C (n6= 1) 2. \int \sec^n x \tan x \ dx = \frac{1}{n} \sec^n x , n\ne 0 {_2F_1}\left[ 1+\frac{a}{2b},1,2+\frac{a}{2b}, -e^{2bx}\right] }& \\ \end{equation}, \begin{equation} \frac{1}{a^2 + b^2} \left[ Forms containing trigonometric functions. \end{equation}, \begin{equation} \int x \sqrt{x^2 \pm a^2}\ dx= \frac{1}{3}\left ( x^2 \pm a^2 \right)^{3/2} Table of Basic Integrals Basic Forms 1 Z (1) xn dx = xn+1 , n 6= −1 n+1 1 Z (2) dx = ln |x| x Z Z (3) udv = uv − vdu 1 1 Z (4) dx = ln |ax + b| ax + b a Integrals of Rational Functions 1 1 Z (5) 2 dx = − (x + a) x+a (x + a)n+1 Z n (6) (x + a) dx = , n 6= −1 n+1 (x + a)n+1 ((n + 1)x − a) Z (7) x(x + a)n dx = (n + 1)(n + 2) 1 Z (8) dx = tan−1 x 1 + x2 1 1 Z −1 x (9) dx = tan a2 + x2 a a 1 1 Z x (10) dx = ln |a2 + x2 | a2 +x 2 2 \end{equation}, \begin{equation} The clustrmap is Page 13/24. b \cosh bx \sin ax – \int x^3 e^{x}\ dx = \left(x^3-3x^2 + 6x – 6\right) e^{x} 13. \end{equation}, \begin{equation} Free math lessons and math homework help from basic math to algebra, geometry and beyond. таблица интегралов. 100. [latex]\int {u}^{n}du=\frac{{u}^{n+1}}{n+1}+C,n\ne \text{−}1[/latex] 2. \int (\ln x)^3\ dx = -6 x+x (\ln x)^3-3 x (\ln x)^2+6 x \ln x \\ \frac{2}{15}(2a+3x)(x-a)^{3/2} \int x \ln x \ dx = \frac{1}{2} x^2 \ln x-\frac{x^2}{4} \int x^n \ln x\ dx = x^{n+1}\left( \dfrac{\ln x}{n+1}-\dfrac{1}{(n+1)^2}\right),\hspace{2ex} n\neq -1 \end{equation}, \begin{equation}\label{eq:Duley} \int \sin^2 ax \cos^2 bx dx = \frac{x}{4} Table of Integrals BASIC FORMS (1)!xndx= 1 n+1 xn+1 (2) 1 x!dx=lnx (3)!udv=uv"!vdu (4) "u(x)v!(x)dx=u(x)v(x)#"v(x)u! \end{equation}, \begin{equation} \int \frac{1}{ax^2+bx+c}dx = \frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} On this page, the tables contain examples of the most common integrals. This page lists some of the most common antiderivatives. -\frac{1}{2}(i)^{n+1}\left [ \Gamma(n+1, -ix) \end{equation}, \begin{equation} \text{erf}\left(i\sqrt{ax}\right), The following is a list of integrals (antiderivative functions) of trigonometric functions.For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions.For a complete list of antiderivative functions, see Lists of integrals.For the special antiderivatives involving trigonometric functions, see Trigonometric integral. \displaystyle{ \frac{\cos[(a+b)x]}{2(a+b)} , a\ne b = uv " vdu ! \text{ where erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt \int \sec x \tan x\ dx = \sec x \int \frac{1}{\sqrt{x\pm a}}\ dx = 2\sqrt{x\pm a} \end{equation}, \begin{equation}\label{eq:swift2} \int x^n \cos ax \ dx = \end{equation}, \begin{equation} ���_eE�j��M���X{�x��4�×oJ����@��p8S9<>$oo�U���{�LrR뾉�눖����E�9OYԚ�X����E��\��� �k�o�r�f�Y��#�j�:�#�x��sƉ�&��R�w��Aj��Dq�d���1t�P����B�wC�D�(ɓ�f�H�"�Ț�`��HĔ� ���r�0�ZN����.�l2����76}�;L���H�� �ᬦ�cRk��ё(c��`+���C�Q�ٙ��tK�eR���9&ׄ�^�X�0l���9��HjNC��Dxԗ)�%tzw��8�u9dKB*��>\�+�. -\frac{1}{a}{\cos ax} \hspace{2mm}{_2F_1}\left[ (A) The Power Rule : Examples : d dx {un} = nu n−1. \end{equation}, \begin{equation} \int \sqrt{x} e^{ax}\ dx = \frac{1}{a}\sqrt{x}e^{ax} \\ & \left. Integral Table. \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:ajoy} (x+a)ndx=(x+a)n a 1+n + x 1+n " #$ % &', n! \frac{2 a}{3} \left({x-a}\right)^{3/2} +\frac{2 }{5}\left( {x-a}\right)^{5/2},\text{ or} Table of Integrals BASIC FORMS (1) x n dx ! -\frac{\sin[(2a-b)x]}{4(2a-b)} \int \frac{1}{a^2+x^2}dx = \frac{1}{a}\tan^{-1}\frac{x}{a} \int x e^x \sin x\ dx = \frac{1}{2}e^x (\cos x – x \cos x + x \sin x) \end{equation}, \begin{equation}\label{eq:Winokur2} \end{equation}, \begin{equation}\label{eq:swift3} \int \ln ( x^2 – a^2 )\hspace{.5ex} {dx} = x \ln (x^2 – a^2 ) +a\ln \frac{x+a}{x-a} – 2x \int x \tan^2 x \ dx = -\frac{x^2}{2} + \ln \cos x + x \tan x Table of Trig Integrals. \end{equation}, \begin{equation} \int \tan^2 ax\ dx = -x + \frac{1}{a} \tan ax \int \cos ax \sinh bx\ dx = Free Integration Worksheet. \end{equation}, \begin{equation} +\frac{1}{2}a^2\tan^{-1}\frac{x}{\sqrt{a^2-x^2}} \int \frac{1}{1+x^2}dx = \tan^{-1}x b \cosh bx \sinh ax \end{equation}, \begin{equation} \end{equation}, \begin{equation} {_2F_1}\left[ \end{equation}, \begin{equation} Sometimes restrictions need to be placed on the values of some of the variables. [latex]\int \frac{du}{u}=\text{ln}|u|+C[/latex] 3. 112. \int x^2 \cos x \ dx = 2 x \cos x + \left ( x^2 – 2 \right ) \sin x 107. \frac{2}{15 a^2}(-2b^2+abx + 3 a^2 x^2) \frac{1}{2}(ia)^{1-n}\left [ (-1)^n \Gamma(n+1, -iax) 2. \int e^x \cos x\ dx = \frac{1}{2}e^x (\sin x + \cos x) \end{equation}, \begin{equation} \int u dv = uv – \int v du 110. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \frac{1}{x}dx = \ln |x| \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int (\ln x)^2\ dx = 2x – 2x \ln x + x (\ln x)^2 x��ZIs�F��W�V�v�KR9$�qj\SS5c�e�h 2 \sqrt{a} \sqrt{ax^2+bx+c} \end{equation}, \begin{equation} \int x\sqrt{x-a}\ dx = \int \cos^2 ax\ dx = \frac{x}{2}+\frac{ \sin 2ax}{4a} The following is a table of formulas of the commonly used Indefinite Integrals. \int e^{bx} \sin ax\ dx = \frac{1}{a^2+b^2}e^{bx} (b\sin ax – a\cos ax) \end{equation}, \begin{equation}\label{eq:ritzert} – \sin x + x \sin x) \frac{1}{2}\left( x^2 – \frac{a^2}{b^2} \right ) \ln \left (a^2 -b^2 x^2 \right) \int \sqrt{\frac{x}{a+x}}\ dx = \sqrt{x(a+x)} \end{equation}, \begin{equation} b \sin ax \sinh bx \int e^{bx} \cos ax\ dx = \frac{1}{a^2 + b^2} e^{bx} ( a \sin ax + b \cos ax ) \int\sqrt{x^2 \pm a^2}\ dx = \frac{1}{2}x\sqrt{x^2\pm a^2} & a\ne b \\ \int (x+a)^n dx = \frac{(x+a)^{n+1}}{n+1}, n\ne -1 \end{equation}, \begin{equation} \int x^2 \cos ax \ dx = \frac{2 x \cos ax }{a^2} + \frac{ a^2 x^2 – 2 }{a^3} \sin ax 4)>$�ÿ�K��1��~)���$��z!~Z��dBPb�H2͈к$��*��'�z�E���D�S#J���t�u�aլM��$.1�����8Q���q3Ds�d-���YOeU)(h��$ �Dp�XBm� } + (-1)^n \Gamma(n+1, ix)\right] Table of integrals - the basic formulas of indefinite integrals.Formulas:- Basic Integration Formulas- Integrals of the rational functions of part- Integrals of transcendental functions- Integrals of the irrational functions of part- Integrals of trigonometric functions of part- Property of indeterminate integrals- Properties of the Definite Integral \int \ln (ax + b) \ dx = \left ( x + \frac{b}{a} \right) \ln (ax+b) – x , a\ne 0 Apr 30, 2018 - Complete table of integrals in a single sheet. \right] It is essential for mathematicians, scientists, and engineers, who rely on it when identifying and subsequently solving extremely complex problems. \frac{1}{\sqrt{a}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right | \int x e^{-ax^2}\ {dx} = -\dfrac{1}{2a}e^{-ax^2} 108. \frac{b^2}{8a^2x}+ Table of Integrals. stream \int x(x+a)^n dx = \frac{(x+a)^{n+1} ( (n+1)x-a)}{(n+1)(n+2)} Integration is the basic operation in integral calculus. \int x e^x\ dx = (x-1) e^x \right] \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \sqrt{x-a}\ dx = \frac{2}{3}(x-a)^{3/2} 34. \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:yates} +\frac{i\sqrt{\pi}}{2a^{3/2}} \int \sin^2 ax \cos bx\ dx = \int x^n e^{ax}\ dx = \frac{(-1)^n}{a^{n+1}}\Gamma[1+n,-ax], \frac{1}{a^2 + b^2} \left[ 104. -2ax + \sinh 2ax \right] As an arbitrary integration constant, the number C, which can be determined if the value of the integral is known at some point.Each function has an infinite number of antiderivatives. \end{equation}, \begin{equation} -\frac{b}{a\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} – = ln x (3) udv ! Not to mention their servers gave up the ghost turned into Zombies on 25 March 2015 (Brains! 4. \int \sin ax \ dx = -\frac{1}{a} \cos ax It is a compilation of the most commonly used integrals. \end{equation}, \begin{equation} \end{array} \int x \cos^2 x \ dx = \frac{x^2}{4}+\frac{1}{8}\cos 2x + \frac{1}{4} x \sin 2x \frac{1}{a^2 + b^2} \left[ \int \frac{1}{\sqrt{a-x}}\ dx = -2\sqrt{a-x} \int \sec x \csc x \ dx = \ln | \tan x | \int x \sec^2 x \ dx = \ln \cos x + x \tan x \int x \cos ax \ dx = \frac{1}{a^2} \cos ax + \frac{x}{a} \sin ax \end{equation}, \begin{equation} \int x e^x \cos x\ dx = \frac{1}{2}e^x (x \cos x \end{equation}, \begin{equation} \left( – 3b^2 + 2 abx + 8 a(c+ax^2) \right) \end{equation}, \begin{equation}\label{eq:Gilmore} \int \frac{x^2}{a^2+x^2}dx = x-a\tan^{-1}\frac{x}{a} \int \tan ax\ dx = -\frac{1}{a} \ln \cos ax \sqrt{x^3(ax+b)} + \int \frac{1}{(x+a)(x+b)}dx = \frac{1}{b-a}\ln\frac{a+x}{b+x}, \text{ } a\ne b Basic Forms Z xndx = 1 n +1 xn+1(1) Z 1 x dx =ln|x| (2) Z udv = uv Z vdu (3) Z 1 ax + b dx = 1 a ln|ax + b| (4) Integrals of Rational Functions Z 1 (x + a)2. dx = 1 x + a (5) Z (x + a)ndx = (x + a)n+1. \frac{1+p}{2}, \frac{1}{2}, \frac{3+p}{2}, \cos^2 ax \int \sqrt{a^2 – x^2}\ dx = \frac{1}{2} x \sqrt{a^2-x^2} \int \frac{x^2}{\sqrt{x^2 \pm a^2}}\ dx = \frac{1}{2}x\sqrt{x^2 \pm a^2} \int \frac{x}{\sqrt{x\pm a} } \ dx = \frac{2}{3}(x\mp 2a)\sqrt{x\pm a} \int \frac{\ln ax}{x}\ dx = \frac{1}{2}\left ( \ln ax \right)^2 \end{equation}, \begin{equation} \begin{cases} \int x^2 e^{x}\ dx = \left(x^2 – 2x + 2\right) e^{x} \hspace{1cm}-\frac{1}{a}e^{ax}{_2F_1}\left[ 1, \frac{a}{2b},1+\frac{a}{2b}, -e^{2bx}\right] \int \frac{1}{\sqrt{a^2 – x^2}}\ dx = \sin^{-1}\frac{x}{a} \end{equation}, \begin{equation}\label{eq:swift1} \end{equation}, \begin{equation} \int x^n e^{ax}\ dx = \dfrac{x^n e^{ax}}{a} – \end{equation}, \begin{equation} \dfrac{n}{a}\int x^{n-1}e^{ax}\hspace{1pt}\text{d}x \end{equation}, \begin{equation} \frac{1}{2}, \frac{1-n}{2}, \frac{3}{2}, \cos^2 ax \end{equation}, \begin{equation} \end{equation}, \begin{equation} a \sin ax \cosh bx + b \cos ax \sinh bx \end{equation}, \begin{equation}\label{eq:Rigo} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int x^n dx = \frac{1}{n+1}x^{n+1},\hspace{1ex}n\neq -1 \int\frac{1}{\sqrt{ax^2+bx+c}}\ dx= 31. \mp \frac{1}{2}a^2 \ln \left| x + \sqrt{x^2\pm a^2} \right | – a \cosh ax \sinh bx \right] \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \sin^3 ax \ dx = -\frac{3 \cos ax}{4a} + \frac{\cos 3ax} {12a} 111. \end{equation}, \begin{equation}\label{eq:xul} \end{split} \end{equation}, \begin{equation} \int \frac{x}{(x+a)^2}dx = \frac{a}{a+x}+\ln |a+x| \end{equation}, \begin{equation} Table of integrals - the basic formulas of indefinite integrals. "1 (8)!x(x+a)ndx= (x+a)1+n(nx+x"a) (n+2)(n+1) (9) dx!1+x2 =tan"1x (10) dx!a2+x2 = 1 a tan"1(x/a) (11) xdx!a2+x2 = 1 2 ln(a2+x2) (12) x2dx!a2+x2 … \end{equation}, \begin{equation}\label{eq:Kloeppel} \int \sec^2 ax\ dx = \frac{1}{a} \tan ax + \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ -b \cosh bx + a \sinh bx ] & a\ne b \\ \int \frac{\ln x}{x^2}\ dx = -\frac{1}{x}-\frac{\ln x}{x} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \sin^2 ax \cos^2 ax\ dx = \frac{x}{8}-\frac{\sin 4ax}{32a} \end{equation}, \begin{equation} 10. \end{equation}, \begin{equation} \int \csc^nx \cot x\ dx = -\frac{1}{n}\csc^n x, n\ne 0 – \frac{\cos bx}{2b} While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{cases} \int \frac{x}{a^2+x^2}dx = \frac{1}{2}\ln|a^2+x^2| \displaystyle{ \frac{ e^{(a+2b)x}}{(a+2b)} \int x \ln (ax + b)\ dx = \frac{bx}{2a}-\frac{1}{4}x^2 Notify me of follow-up comments by email. \int \sin^2 x \cos x\ dx = \frac{1}{3} \sin^3 x \int \cosh ax\ dx =\frac{1}{a} \sinh ax \frac{b^3}{8a^{5/2}}\ln \left | a\sqrt{x} + \sqrt{a(ax+b)} \right | \int \frac{1}{\sqrt{x^2 \pm a^2}}\ dx = \ln \left | x + \sqrt{x^2 \pm a^2} \right | \frac{b}{2a^{3/2}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right | \frac{x}{3}\right] Basic Differentiation Rules Basic Integration Formulas DERIVATIVES AND INTEGRALS © Houghton Mifflin Company, Inc. 1. Table of Integrals Basic Forms xn dx = 1 xn+1 n+1 1 dx = ln |x| x udv = uv vdu 1 1 dx = ln |ax + b| ax + b a (1) (2) These restrictions are shown in the third column. \end{equation}, \begin{equation} 1. ∫ u n d u = u n + 1 n + 1 + C, n ≠ − 1 ∫ u n d u = u n + 1 n + 1 + C, n ≠ − 1. \int \sec x \ dx = \ln | \sec x + \tan x | = 2 \tanh^{-1} \left (\tan \frac{x}{2} \right) \int x^2 \ln x \ dx = \frac{1}{3} x^3 \ln x-\frac{x^3}{9} It includes: Table of Basic Forms; Table of Rational Integrals; Table of Integrals with Roots; Table of Integrals with Logarithms; Table of Exponential Integrals; Table of Trigonometric Integrals Forms … \int e^{ax^2}\ dx = -\frac{i\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(ix\sqrt{a}\right) Integration — is one of the main mathematical operations. \end{equation}, \begin{equation} \text{ where } \Gamma(a,x)=\int_x^{\infty} t^{a-1}e^{-t}\hspace{2pt}\text{d}t \int \ln ax\ dx = x \ln ax – x Administrator of Mini Physics. \int \tan^3 ax dx = \frac{1}{a} \ln \cos ax + \frac{1}{2a}\sec^2 ax 99. a \sin ax \sinh bx Students, teachers, parents, and everyone can find solutions to their math problems instantly. \frac{1}{a}\sqrt{ax^2+bx + c} \begin{array}{l} +\frac{1}{2}\left(x^2-\frac{b^2}{a^2}\right)\ln (ax+b) \end{equation}, \begin{equation} \\ \frac{2}{3} x(x-a)^{3/2} – \frac{4}{15} (x-a)^{5/2}, \text{ or} \end{equation}, \begin{equation} \int \csc^2 ax\ dx = -\frac{1}{a} \cot ax – \frac{\sin[(2a+b)x]}{4(2a+b)} \int \frac{x}{\sqrt{a^2-x^2}}\ dx = -\sqrt{a^2-x^2} + \frac{\sin bx}{2b} \end{equation}, \begin{equation} Use the table of integral formulas and the rules above to evaluate the following integrals. Table of Integrals Engineers usually refer to a table of integrals when performing calculations involving integration. \int x \sin^2 x \ dx = \frac{x^2}{4}-\frac{1}{8}\cos 2x – \frac{1}{4} x \sin 2x a \cos ax \sinh bx \int e^{-ax^2}\ dx = \frac{\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(x\sqrt{a}\right) \int\frac{dx}{(a^2+x^2)^{3/2}}=\frac{x}{a^2\sqrt{a^2+x^2}} Table of Standard Integrals 1. \end{equation}, \begin{equation} \int &x \sqrt{a x^2 + bx + c}\ dx = \frac{1}{48a^{5/2}}\left ( \int \sqrt{\frac{x}{a-x}}\ dx = -\sqrt{x(a-x)} \end{equation}, \begin{equation} \int e^{ax}\ dx = \frac{1}{a}e^{ax} 16. Morganton, Ga Weather, Can You Use Acrylic Paint On Watercolor Paper, Wellesley Chemistry Faculty, Corgi Hdb Appeal, Desert Marigold Edible, Adding Decimals Worksheet, Classic Infrared Heater And Air Purifier, Wine And Roses Weigela Evergreen, Uss Knox Reunion, Malayalam Movie Queen, " />
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\int x^2 e^{-ax^2}\ {dx} = \dfrac{1}{4}\sqrt{\dfrac{\pi}{a^3}}\text{erf}(x\sqrt{a}) -\dfrac{x}{2a}e^{-ax^2} All the immediate integrals. \end{equation}, \begin{equation} \int \sqrt{a x^2 + b x + c}\ dx = 22. Example: \displaystyle{\frac{e^{ax}-2\tan^{-1}[e^{ax}]}{a} } & a = b \frac{\sin[2(a-b)x]}{16(a-b)} \end{equation}, \begin{equation} \int x^n \cos x dx = 98. (x)dx RATIONAL FUNCTIONS (5) 1 ax+b!dx= 1 a ln(ax+b) (6) 1 (x+a)2!dx= "1 x+a (7)! \end{equation}, \begin{equation} \frac{1}{b^2-a^2}\left[ \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:Winokur1} 7. Made with | 2010 - 2020 | Mini Physics |, Click to share on Twitter (Opens in new window), Click to share on Facebook (Opens in new window), Click to share on Reddit (Opens in new window), Click to share on Telegram (Opens in new window), Click to share on WhatsApp (Opens in new window), Click to share on LinkedIn (Opens in new window), Click to share on Tumblr (Opens in new window), Click to share on Pinterest (Opens in new window), Click to share on Pocket (Opens in new window), Click to share on Skype (Opens in new window), Mathematics For An Undergraduate Physics Course, Case Study 2: Energy Conversion for A Bouncing Ball, Case Study 1: Energy Conversion for An Oscillating Ideal Pendulum, Practice MCQs For Measurement of Physical Quantities, O Level: Magnetic Field And Magnetic Field Lines. \frac{1}{a^2 + b^2} \left[ \int \cos x \sin x\ dx = \frac{1}{2}\sin^2 x + c_1 = -\frac{1}{2} \cos^2x + c_2 = -\frac{1}{4} \cos 2x + c_3 Table of Indefinite Integral Formulas . \int x^2 (\ln x)^2\ dx = \frac{2 x^3}{27}+\frac{1}{3} x^3 (\ln x)^2-\frac{2}{9} x^3 \ln x 7 0 obj << \int \cos^p ax dx = -\frac{1}{a(1+p)}{\cos^{1+p} ax} \times \int \ln \left ( ax^2 + bx + c\right) \ dx = \frac{1}{a}\sqrt{4ac-b^2}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} \int \tan^n ax\ dx = \int \sin ax \cosh bx \ dx = \int \cos^3 ax dx = \frac{3 \sin ax}{4a}+\frac{ \sin 3ax}{12a} \end{equation}, \begin{equation} 101. -a\tan^{-1}\frac{\sqrt{x(a-x)}}{x-a} ��H�$e���׍� �XH*N�"���뷿�u7M>$4��������kffgJ&��N9�N'�jB�Mn�ۅ����C�ȄQ��}����n�*��Y�����a����� � \int \sin ax \sinh bx \ dx = \right. \end{equation}, \begin{equation} \end{equation}, \begin{equation} Table of Basic Integrals Basic Forms There have been visitors to integral-table.com since 2004. \end{equation}, \begin{equation} -2x + \left( \frac{b}{2a}+x \right )\ln \left (ax^2+bx+c \right) You can verify any of the formulas by differentiating the function on the right side and obtaining the integrand. \int \ln ( x^2 + a^2 )\hspace{.5ex} {dx} = x \ln (x^2 + a^2 ) +2a\tan^{-1} \frac{x}{a} – 2x \right] \right] \end{equation}, \begin{equation} \int x e^{ax}\ dx = \left(\frac{x}{a}-\frac{1}{a^2}\right) e^{ax} (3x2 + 4)d dx {u} = 12 u.u d dx { 2 − 4x2 + 7x5} = 1 2 2 − 4x2 + 7x5 (−8x + 35x4) d dx {c} = 0 , c is a constant ddx {6} = 0 , since ≅ 3.14 is a constant. \int \cos^2 ax \sin bx\ dx = \frac{\cos[(2a-b)x]}{4(2a-b)} \frac{\tan^{n+1} ax }{a(1+n)} \times \int e^x \sin x \ dx = \frac{1}{2}e^x (\sin x – \cos x) -b^2 \ln \left| a\sqrt{x} + \sqrt{a(ax+b)} \right| \right ] \end{equation}, \begin{equation} {_2}F_1\left( \frac{n+1}{2}, \right] The table presents a selection of integrals found in the Calculus books. \end{equation}, \begin{equation} Indefinite integrals. %PDF-1.5 Table of Basic Integrals1 (1) Z xndx = 1 n+1 xn+1; n 6= 1 (2) Z 1 x dx = lnjxj (3) Z u dv = uv Z vdu (4) Z exdx = e (5) Z axdx = 1 lna ax (6) Z lnxdx = xlnx x (7) Z sinxdx = cosx (8) Z cosxdx = sinx (9) Z tanxdx = lnjsecxj (10) Z secxdx = lnjsecx+tanxj (11) Z sec2xdx = tanx (12) Z secxtanxdx = secx (13) Z a a2+x2 dx = tan1 x a (14) Z a a2x2 1. \end{equation}, \begin{equation} \int x^2 \sin x\ dx = \left(2-x^2\right) \cos x + 2 x \sin x \frac{b+2ax}{4a}\sqrt{ax^2+bx+c} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int e^{ax} \cosh bx \ dx = \end{equation}, \begin{equation}\label{eq:ebke} \int \csc^3 x\ dx = -\frac{1}{2}\cot x \csc x + \frac{1}{2} \ln | \csc x – \cot x | \int \frac{x}{\sqrt{ax^2+bx+c}}\ dx= 113. \int x \sin x\ dx = -x \cos x + \sin x \end{cases} \end{equation}, \begin{equation} Basic Integrals. Forms containing logarithms and exponentials. \int \sqrt{ax+b}\ dx = \left(\frac{2b}{3a}+\frac{2x}{3}\right)\sqrt{ax+b} Mini Physics is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon.sg. \int \sinh ax \cosh ax dx= \end{equation}, \begin{equation} \end{equation}, \begin{equation} \frac{\sin[2(a+b)x]}{16(a+b)} \end{equation}, \begin{equation}\label{eq:dewitt} \displaystyle{\frac{e^{2ax}}{4a} – \frac{x}{2}} & a = b \int \frac{1}{ax+b}dx = \frac{1}{a} \ln |ax + b| \end{equation}, \begin{equation}\label{eq:veky} \pm\frac{1}{2}a^2 \ln \left | x + \sqrt{x^2\pm a^2} \right | \int \sqrt{x(ax+b)}\ dx = \frac{1}{4a^{3/2}}\left[(2ax + b)\sqrt{ax(ax+b)} 28. /Length 2403 \end{equation}, \begin{equation} Integral tables >> Basic forms. \begin{cases} 1. 109. This leaflet provides such a table. 105. \begin{cases} Table Of Basic Integrals Basic Forms \begin{equation} \int x^n dx = \frac{1}{n+1}x^{n+1},\hspace{1ex}n\neq -1 \end{equation} \begin{equation} \int \frac{1}{x}dx = \ln |x| \int \frac{x}{\sqrt{x^2\pm a^2}}\ dx = \sqrt{x^2 \pm a^2} \end{equation}, \begin{equation} \frac{1}{4a}\left[ -a \cos ax \cosh bx + \int \cos ax \sin bx\ dx = \frac{\cos[(a-b) x]}{2(a-b)} – Forms containing inverse trigonometric functions. \end{equation}, $$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2}} = \sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+ \infty}_{- \infty}x^{2n} e^{-ax^{2}} = (-1)^{n} \frac{\partial^{n}}{\partial a^{n}}\sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2} + bx} = e^{\frac{b^2}{4a}}\sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \sin^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 – \frac{6(-1)^n}{n^2 \pi^2} \right)$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \cos^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 + \frac{6(-1)^n}{n^2 \pi^2} \right)$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x \cos \left( \frac{ \pi x}{a} \right) \sin \left( \frac{2 \pi x}{a} \right) = \frac{8a^2}{9 \pi ^2} $$, $$\int\limits^{a}_{b} \frac{dx}{\sqrt{\left(a-x \right) \left(x-b \right)}} = \pi \text{ for a > b}$$, $$\int\limits^{a}_{b} \frac{dx}{x\sqrt{\left(a-x \right) \left(x-b \right)}} = \frac{ \pi}{\sqrt{ab}} \text{ for a > b > 0}$$, $$\int\limits^{\frac{\pi}{2}}_{- \frac{\pi}{2}} \frac{dx}{1+ y \sin x} = \frac{\pi}{\sqrt{1 – y^2}} \text{ for -1 < y < 1}$$, $$\int \frac{dx}{\sqrt{a^{2} – x^{2}}} = \text{arcsin} \, \frac{x}{a}$$, $$\int \frac{x dx}{\sqrt{a^{2} + x^{2}}} = \sqrt{a^{2} + x^{2}}$$, $$\int \frac{dx}{\sqrt{a^{2} +x^{2}}} = \text{ln} \, \left(x + \sqrt{a^{2} + x^{2}} \right)$$, $$\int \frac{dx}{a^{2} +x^{2}} = \frac{1}{a} \, \text{arctan} \, \frac{x}{a}$$, $$\int \frac{dx}{ \left( a^{2} + x^{2} \right)^{\frac{3}{2}}} = \frac{1}{a^{2}} \frac{x}{\sqrt{a^{2} +x^{2}}}$$, $$\int\frac{x \, dx}{ \left( a^{2}+x^{2} \right)^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{a^{2} + x^{2}}}$$, $$\int \frac{dx}{\sqrt{ (x – a)^{2} + b^{2}}} = \text{ln} \, \frac{1}{(a – x) + \sqrt{(a-x)^{2} + b^{2}}}$$, $$\int \frac{(x – a) \, dx}{\left[ (x-a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{(x-a)^{2} + b^{2}}}$$, $$\int \frac{dx}{\left[ (x – a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \frac{x – a}{b^{2} \sqrt{(x – a)^{2} +b^{2}}}$$. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \sqrt{ax+b} \end{equation}, \begin{equation} \end{equation}, \begin{equation} u ddx {(x3 + 4x + 1)3/4} = 34 (x3 + 4x + 1)−1/4. \end{equation}, \begin{equation} [Note that you may need to use more than one of the above rules for one integral]. \end{equation}, \begin{equation} The clustrmap is periodically (and automatically) archived and its counters reset, so the total is smaller. \int \sqrt{x^3(ax+b)} \ dx =\left [ \end{equation}, \begin{equation} Types of Integrals. Basic forms. \end{equation}, \begin{equation} \end{equation}, \begin{equation} There have been visitors to integral-table.com since 2004. \end{equation}, \begin{equation}\label{eq:Larry-Morris}\begin{split} -\Gamma(n+1, ixa)\right] \int \sinh ax\ dx = \frac{1}{a} \cosh ax \end{equation}, \begin{equation} \end{equation}, \begin{equation} 103. Tinycards by Duolingo is a fun flashcard app that helps you memorize anything for free, forever. \int \cos ax \cosh bx\ dx = \int \sec^3 x \ {dx} = \frac{1}{2} \sec x \tan x + \frac{1}{2}\ln | \sec x + \tan x | \end{equation}, \begin{equation} \end{equation}, \begin{equation} Not to mention their servers \end{equation}, \begin{equation} \displaystyle{\frac{e^{2ax}}{4a} + \frac{x}{2}} & a = b – \frac{\cos[(2a+b)x]}{4(2a+b)} \end{equation}, \begin{equation} Table of Integrals∗. \right] If you spot any errors or want to suggest improvements, please contact us. \right . 19. = 1 n + 1 x n + 1 (2) 1 x dx ! \end{equation}, \begin{equation} \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ a \cosh bx – b \sinh bx ] & a\ne b \\ \int \csc x\ dx = \ln \left | \tan \frac{x}{2} \right| = \ln | \csc x – \cot x| + C The Table of Integrals, Series, and Products is the major reference source for integrals in the English language. Note: Most of the following integral entries are written for indefinite integrals, but they also apply to definite integrals. \int e^{ax} \sinh bx \ dx = \frac{b}{12a}- \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int x \sin ax\ dx = -\frac{x \cos ax}{a} + \frac{\sin ax}{a^2} 102. + 3(b^3-4abc)\ln \left|b + 2ax + 2\sqrt{a}\sqrt{ax^2+bx+c} \right| \right) \int x \ln \left ( a^2 – b^2 x^2 \right )\ dx = -\frac{1}{2}x^2+ \int \tanh ax\hspace{1.5pt} dx =\frac{1}{a} \ln \cosh ax \int x \sqrt{ax + b}\ dx = \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} -a\ln \left [ \sqrt{x} + \sqrt{x+a}\right] \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:Russ} b \cos ax \cosh bx + \end{equation}, \begin{equation} \end{equation}, \begin{equation} Read Free Table Of Integrals Integral Table periodically (and automatically) archived and its counters reset, so the total is smaller. +\frac{\sin 2bx}{8b}- \int x \cos x \ dx = \cos x + x \sin x \left\{ \end{equation}, \begin{equation} \int x^2 \sin ax\ dx =\frac{2-a^2x^2}{a^3}\cos ax +\frac{ 2 x \sin ax}{a^2} \int \sec^2 x \tan x\ dx = \frac{1}{2} \sec^2 x \end{cases} \int \frac{1}{(x+a)^2}dx = -\frac{1}{x+a} Free Table of Integrals to print on a single sheet side and side. \int \cos ax\ dx= \frac{1}{a} \sin ax – (-1)^n\Gamma(n+1, -ix)\right] \end{equation}, \begin{equation} \end{equation}, \begin{equation} View Notes - Table_of_Integrals from MAP 3305 at Florida Atlantic University. \end{equation}, \begin{equation} Scroll down the page if you need more examples and step by step solutions of indefinite integrals. >> \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:qarles1} Basic Integrals; Trigonometric Integrals; Exponential and Logarithmic Integrals; Hyperbolic Integrals; Inverse Trigonometric Integrals; Integrals Involving a2 + u2, a > 0; Integrals Involving u2 − a2, a > 0; Integrals Involving a2 − u2, a > 0; Integrals Involving 2au − u2, a > 0; Integrals … \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \sin^n ax \ dx = \end{equation}, \begin{equation} %���� A: TABLE OF BASIC DERIVATIVES Let u = u(x) be a differentiable function of the independent variable x, that is u(x) exists. 1, \frac{n+3}{2}, -\tan^2 ax \right) \int \sinh ax \cosh bx \ dx = \int \cos^2 ax \sin ax\ dx = -\frac{1}{3a}\cos^3{ax} \int x (\ln x)^2\ dx = \frac{x^2}{4}+\frac{1}{2} x^2 (\ln x)^2-\frac{1}{2} x^2 \ln x -\frac{\sin 2ax}{8a}- \end{equation}, \begin{equation} For the following, the letters a, b, n, and C represent constants.. \int \frac{x^3}{a^2+x^2}dx = \frac{1}{2}x^2-\frac{1}{2}a^2\ln|a^2+x^2| \end{equation}, \begin{equation}\label{eq:Weems} \end{equation}, \begin{equation} \int \sin^2 ax\ dx = \frac{x}{2} – \frac{\sin 2ax} {4a} \end{equation}, \begin{equation} /Filter /FlateDecode \end{equation}, \begin{equation} \int \frac{x}{ax^2+bx+c}dx = \frac{1}{2a}\ln|ax^2+bx+c| Integrals Involving a + bu, a ≠ 0. 1. ∫ (1 / 2) ln (x) dx 2. ∫ [sin (x) + x 5] dx 3. ∫ [sinh (x) - 3] dx 4. ∫ - x sin (x) dx 5. \int e^{ax} \tanh bx\ dx = \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:qarles2} Basic Integrals. \int x^2 e^{ax}\ dx = \left(\frac{x^2}{a}-\frac{2x}{a^2}+\frac{2}{a^3}\right) e^{ax} \frac{4ac-b^2}{8a^{3/2}}\ln \left| 2ax + b + 2\sqrt{a(ax^2+bx^+c)}\right | \int x^n \sin x \ dx = -\frac{1}{2}(i)^n\left[ \Gamma(n+1, -ix) These tinycards help you memorize the table of basic integrals. \int (ax+b)^{3/2}\ dx =\frac{2}{5a}(ax+b)^{5/2} Formulas: - Basic Integration Formulas - Integrals of the rational functions of part - Integrals of transcendental functions - Integrals of the irrational functions of part - Integrals of trigonometric functions of part - Property of indeterminate integrals - Properties of the Definite Integral 25. Home University Mathematics Integration Table, \begin{equation} 106. Z xndx= xn+1 n+1 +C (n6= 1) 2. \int \sec^n x \tan x \ dx = \frac{1}{n} \sec^n x , n\ne 0 {_2F_1}\left[ 1+\frac{a}{2b},1,2+\frac{a}{2b}, -e^{2bx}\right] }& \\ \end{equation}, \begin{equation} \frac{1}{a^2 + b^2} \left[ Forms containing trigonometric functions. \end{equation}, \begin{equation} \int x \sqrt{x^2 \pm a^2}\ dx= \frac{1}{3}\left ( x^2 \pm a^2 \right)^{3/2} Table of Basic Integrals Basic Forms 1 Z (1) xn dx = xn+1 , n 6= −1 n+1 1 Z (2) dx = ln |x| x Z Z (3) udv = uv − vdu 1 1 Z (4) dx = ln |ax + b| ax + b a Integrals of Rational Functions 1 1 Z (5) 2 dx = − (x + a) x+a (x + a)n+1 Z n (6) (x + a) dx = , n 6= −1 n+1 (x + a)n+1 ((n + 1)x − a) Z (7) x(x + a)n dx = (n + 1)(n + 2) 1 Z (8) dx = tan−1 x 1 + x2 1 1 Z −1 x (9) dx = tan a2 + x2 a a 1 1 Z x (10) dx = ln |a2 + x2 | a2 +x 2 2 \end{equation}, \begin{equation} The clustrmap is Page 13/24. b \cosh bx \sin ax – \int x^3 e^{x}\ dx = \left(x^3-3x^2 + 6x – 6\right) e^{x} 13. \end{equation}, \begin{equation} Free math lessons and math homework help from basic math to algebra, geometry and beyond. таблица интегралов. 100. [latex]\int {u}^{n}du=\frac{{u}^{n+1}}{n+1}+C,n\ne \text{−}1[/latex] 2. \int (\ln x)^3\ dx = -6 x+x (\ln x)^3-3 x (\ln x)^2+6 x \ln x \\ \frac{2}{15}(2a+3x)(x-a)^{3/2} \int x \ln x \ dx = \frac{1}{2} x^2 \ln x-\frac{x^2}{4} \int x^n \ln x\ dx = x^{n+1}\left( \dfrac{\ln x}{n+1}-\dfrac{1}{(n+1)^2}\right),\hspace{2ex} n\neq -1 \end{equation}, \begin{equation}\label{eq:Duley} \int \sin^2 ax \cos^2 bx dx = \frac{x}{4} Table of Integrals BASIC FORMS (1)!xndx= 1 n+1 xn+1 (2) 1 x!dx=lnx (3)!udv=uv"!vdu (4) "u(x)v!(x)dx=u(x)v(x)#"v(x)u! \end{equation}, \begin{equation} \int \frac{1}{ax^2+bx+c}dx = \frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} On this page, the tables contain examples of the most common integrals. This page lists some of the most common antiderivatives. -\frac{1}{2}(i)^{n+1}\left [ \Gamma(n+1, -ix) \end{equation}, \begin{equation} \text{erf}\left(i\sqrt{ax}\right), The following is a list of integrals (antiderivative functions) of trigonometric functions.For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions.For a complete list of antiderivative functions, see Lists of integrals.For the special antiderivatives involving trigonometric functions, see Trigonometric integral. \displaystyle{ \frac{\cos[(a+b)x]}{2(a+b)} , a\ne b = uv " vdu ! \text{ where erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt \int \sec x \tan x\ dx = \sec x \int \frac{1}{\sqrt{x\pm a}}\ dx = 2\sqrt{x\pm a} \end{equation}, \begin{equation}\label{eq:swift2} \int x^n \cos ax \ dx = \end{equation}, \begin{equation} ���_eE�j��M���X{�x��4�×oJ����@��p8S9<>$oo�U���{�LrR뾉�눖����E�9OYԚ�X����E��\��� �k�o�r�f�Y��#�j�:�#�x��sƉ�&��R�w��Aj��Dq�d���1t�P����B�wC�D�(ɓ�f�H�"�Ț�`��HĔ� ���r�0�ZN����.�l2����76}�;L���H�� �ᬦ�cRk��ё(c��`+���C�Q�ٙ��tK�eR���9&ׄ�^�X�0l���9��HjNC��Dxԗ)�%tzw��8�u9dKB*��>\�+�. -\frac{1}{a}{\cos ax} \hspace{2mm}{_2F_1}\left[ (A) The Power Rule : Examples : d dx {un} = nu n−1. \end{equation}, \begin{equation} \int \sqrt{x} e^{ax}\ dx = \frac{1}{a}\sqrt{x}e^{ax} \\ & \left. Integral Table. \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:ajoy} (x+a)ndx=(x+a)n a 1+n + x 1+n " #$ % &', n! \frac{2 a}{3} \left({x-a}\right)^{3/2} +\frac{2 }{5}\left( {x-a}\right)^{5/2},\text{ or} Table of Integrals BASIC FORMS (1) x n dx ! -\frac{\sin[(2a-b)x]}{4(2a-b)} \int \frac{1}{a^2+x^2}dx = \frac{1}{a}\tan^{-1}\frac{x}{a} \int x e^x \sin x\ dx = \frac{1}{2}e^x (\cos x – x \cos x + x \sin x) \end{equation}, \begin{equation}\label{eq:Winokur2} \end{equation}, \begin{equation}\label{eq:swift3} \int \ln ( x^2 – a^2 )\hspace{.5ex} {dx} = x \ln (x^2 – a^2 ) +a\ln \frac{x+a}{x-a} – 2x \int x \tan^2 x \ dx = -\frac{x^2}{2} + \ln \cos x + x \tan x Table of Trig Integrals. \end{equation}, \begin{equation} \int \tan^2 ax\ dx = -x + \frac{1}{a} \tan ax \int \cos ax \sinh bx\ dx = Free Integration Worksheet. \end{equation}, \begin{equation} +\frac{1}{2}a^2\tan^{-1}\frac{x}{\sqrt{a^2-x^2}} \int \frac{1}{1+x^2}dx = \tan^{-1}x b \cosh bx \sinh ax \end{equation}, \begin{equation} \end{equation}, \begin{equation} {_2F_1}\left[ \end{equation}, \begin{equation} Sometimes restrictions need to be placed on the values of some of the variables. [latex]\int \frac{du}{u}=\text{ln}|u|+C[/latex] 3. 112. \int x^2 \cos x \ dx = 2 x \cos x + \left ( x^2 – 2 \right ) \sin x 107. \frac{2}{15 a^2}(-2b^2+abx + 3 a^2 x^2) \frac{1}{2}(ia)^{1-n}\left [ (-1)^n \Gamma(n+1, -iax) 2. \int e^x \cos x\ dx = \frac{1}{2}e^x (\sin x + \cos x) \end{equation}, \begin{equation} \int u dv = uv – \int v du 110. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \frac{1}{x}dx = \ln |x| \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int (\ln x)^2\ dx = 2x – 2x \ln x + x (\ln x)^2 x��ZIs�F��W�V�v�KR9$�qj\SS5c�e�h 2 \sqrt{a} \sqrt{ax^2+bx+c} \end{equation}, \begin{equation} \int x\sqrt{x-a}\ dx = \int \cos^2 ax\ dx = \frac{x}{2}+\frac{ \sin 2ax}{4a} The following is a table of formulas of the commonly used Indefinite Integrals. \int e^{bx} \sin ax\ dx = \frac{1}{a^2+b^2}e^{bx} (b\sin ax – a\cos ax) \end{equation}, \begin{equation}\label{eq:ritzert} – \sin x + x \sin x) \frac{1}{2}\left( x^2 – \frac{a^2}{b^2} \right ) \ln \left (a^2 -b^2 x^2 \right) \int \sqrt{\frac{x}{a+x}}\ dx = \sqrt{x(a+x)} \end{equation}, \begin{equation} b \sin ax \sinh bx \int e^{bx} \cos ax\ dx = \frac{1}{a^2 + b^2} e^{bx} ( a \sin ax + b \cos ax ) \int\sqrt{x^2 \pm a^2}\ dx = \frac{1}{2}x\sqrt{x^2\pm a^2} & a\ne b \\ \int (x+a)^n dx = \frac{(x+a)^{n+1}}{n+1}, n\ne -1 \end{equation}, \begin{equation} \int x^2 \cos ax \ dx = \frac{2 x \cos ax }{a^2} + \frac{ a^2 x^2 – 2 }{a^3} \sin ax 4)>$�ÿ�K��1��~)���$��z!~Z��dBPb�H2͈к$��*��'�z�E���D�S#J���t�u�aլM��$.1�����8Q���q3Ds�d-���YOeU)(h��$ �Dp�XBm� } + (-1)^n \Gamma(n+1, ix)\right] Table of integrals - the basic formulas of indefinite integrals.Formulas:- Basic Integration Formulas- Integrals of the rational functions of part- Integrals of transcendental functions- Integrals of the irrational functions of part- Integrals of trigonometric functions of part- Property of indeterminate integrals- Properties of the Definite Integral \int \ln (ax + b) \ dx = \left ( x + \frac{b}{a} \right) \ln (ax+b) – x , a\ne 0 Apr 30, 2018 - Complete table of integrals in a single sheet. \right] It is essential for mathematicians, scientists, and engineers, who rely on it when identifying and subsequently solving extremely complex problems. \frac{1}{\sqrt{a}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right | \int x e^{-ax^2}\ {dx} = -\dfrac{1}{2a}e^{-ax^2} 108. \frac{b^2}{8a^2x}+ Table of Integrals. stream \int x(x+a)^n dx = \frac{(x+a)^{n+1} ( (n+1)x-a)}{(n+1)(n+2)} Integration is the basic operation in integral calculus. \int x e^x\ dx = (x-1) e^x \right] \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \sqrt{x-a}\ dx = \frac{2}{3}(x-a)^{3/2} 34. \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:yates} +\frac{i\sqrt{\pi}}{2a^{3/2}} \int \sin^2 ax \cos bx\ dx = \int x^n e^{ax}\ dx = \frac{(-1)^n}{a^{n+1}}\Gamma[1+n,-ax], \frac{1}{a^2 + b^2} \left[ 104. -2ax + \sinh 2ax \right] As an arbitrary integration constant, the number C, which can be determined if the value of the integral is known at some point.Each function has an infinite number of antiderivatives. \end{equation}, \begin{equation} -\frac{b}{a\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} – = ln x (3) udv ! Not to mention their servers gave up the ghost turned into Zombies on 25 March 2015 (Brains! 4. \int \sin ax \ dx = -\frac{1}{a} \cos ax It is a compilation of the most commonly used integrals. \end{equation}, \begin{equation} \end{array} \int x \cos^2 x \ dx = \frac{x^2}{4}+\frac{1}{8}\cos 2x + \frac{1}{4} x \sin 2x \frac{1}{a^2 + b^2} \left[ \int \frac{1}{\sqrt{a-x}}\ dx = -2\sqrt{a-x} \int \sec x \csc x \ dx = \ln | \tan x | \int x \sec^2 x \ dx = \ln \cos x + x \tan x \int x \cos ax \ dx = \frac{1}{a^2} \cos ax + \frac{x}{a} \sin ax \end{equation}, \begin{equation} \int x e^x \cos x\ dx = \frac{1}{2}e^x (x \cos x \end{equation}, \begin{equation} \left( – 3b^2 + 2 abx + 8 a(c+ax^2) \right) \end{equation}, \begin{equation}\label{eq:Gilmore} \int \frac{x^2}{a^2+x^2}dx = x-a\tan^{-1}\frac{x}{a} \int \tan ax\ dx = -\frac{1}{a} \ln \cos ax \sqrt{x^3(ax+b)} + \int \frac{1}{(x+a)(x+b)}dx = \frac{1}{b-a}\ln\frac{a+x}{b+x}, \text{ } a\ne b Basic Forms Z xndx = 1 n +1 xn+1(1) Z 1 x dx =ln|x| (2) Z udv = uv Z vdu (3) Z 1 ax + b dx = 1 a ln|ax + b| (4) Integrals of Rational Functions Z 1 (x + a)2. dx = 1 x + a (5) Z (x + a)ndx = (x + a)n+1. \frac{1+p}{2}, \frac{1}{2}, \frac{3+p}{2}, \cos^2 ax \int \sqrt{a^2 – x^2}\ dx = \frac{1}{2} x \sqrt{a^2-x^2} \int \frac{x^2}{\sqrt{x^2 \pm a^2}}\ dx = \frac{1}{2}x\sqrt{x^2 \pm a^2} \int \frac{x}{\sqrt{x\pm a} } \ dx = \frac{2}{3}(x\mp 2a)\sqrt{x\pm a} \int \frac{\ln ax}{x}\ dx = \frac{1}{2}\left ( \ln ax \right)^2 \end{equation}, \begin{equation} \begin{cases} \int x^2 e^{x}\ dx = \left(x^2 – 2x + 2\right) e^{x} \hspace{1cm}-\frac{1}{a}e^{ax}{_2F_1}\left[ 1, \frac{a}{2b},1+\frac{a}{2b}, -e^{2bx}\right] \int \frac{1}{\sqrt{a^2 – x^2}}\ dx = \sin^{-1}\frac{x}{a} \end{equation}, \begin{equation}\label{eq:swift1} \end{equation}, \begin{equation} \int x^n e^{ax}\ dx = \dfrac{x^n e^{ax}}{a} – \end{equation}, \begin{equation} \dfrac{n}{a}\int x^{n-1}e^{ax}\hspace{1pt}\text{d}x \end{equation}, \begin{equation} \frac{1}{2}, \frac{1-n}{2}, \frac{3}{2}, \cos^2 ax \end{equation}, \begin{equation} \end{equation}, \begin{equation} a \sin ax \cosh bx + b \cos ax \sinh bx \end{equation}, \begin{equation}\label{eq:Rigo} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int x^n dx = \frac{1}{n+1}x^{n+1},\hspace{1ex}n\neq -1 \int\frac{1}{\sqrt{ax^2+bx+c}}\ dx= 31. \mp \frac{1}{2}a^2 \ln \left| x + \sqrt{x^2\pm a^2} \right | – a \cosh ax \sinh bx \right] \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \sin^3 ax \ dx = -\frac{3 \cos ax}{4a} + \frac{\cos 3ax} {12a} 111. \end{equation}, \begin{equation}\label{eq:xul} \end{split} \end{equation}, \begin{equation} \int \frac{x}{(x+a)^2}dx = \frac{a}{a+x}+\ln |a+x| \end{equation}, \begin{equation} Table of integrals - the basic formulas of indefinite integrals. "1 (8)!x(x+a)ndx= (x+a)1+n(nx+x"a) (n+2)(n+1) (9) dx!1+x2 =tan"1x (10) dx!a2+x2 = 1 a tan"1(x/a) (11) xdx!a2+x2 = 1 2 ln(a2+x2) (12) x2dx!a2+x2 … \end{equation}, \begin{equation}\label{eq:Kloeppel} \int \sec^2 ax\ dx = \frac{1}{a} \tan ax + \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ -b \cosh bx + a \sinh bx ] & a\ne b \\ \int \frac{\ln x}{x^2}\ dx = -\frac{1}{x}-\frac{\ln x}{x} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \sin^2 ax \cos^2 ax\ dx = \frac{x}{8}-\frac{\sin 4ax}{32a} \end{equation}, \begin{equation} 10. \end{equation}, \begin{equation} \int \csc^nx \cot x\ dx = -\frac{1}{n}\csc^n x, n\ne 0 – \frac{\cos bx}{2b} While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{cases} \int \frac{x}{a^2+x^2}dx = \frac{1}{2}\ln|a^2+x^2| \displaystyle{ \frac{ e^{(a+2b)x}}{(a+2b)} \int x \ln (ax + b)\ dx = \frac{bx}{2a}-\frac{1}{4}x^2 Notify me of follow-up comments by email. \int \sin^2 x \cos x\ dx = \frac{1}{3} \sin^3 x \int \cosh ax\ dx =\frac{1}{a} \sinh ax \frac{b^3}{8a^{5/2}}\ln \left | a\sqrt{x} + \sqrt{a(ax+b)} \right | \int \frac{1}{\sqrt{x^2 \pm a^2}}\ dx = \ln \left | x + \sqrt{x^2 \pm a^2} \right | \frac{b}{2a^{3/2}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right | \frac{x}{3}\right] Basic Differentiation Rules Basic Integration Formulas DERIVATIVES AND INTEGRALS © Houghton Mifflin Company, Inc. 1. Table of Integrals Basic Forms xn dx = 1 xn+1 n+1 1 dx = ln |x| x udv = uv vdu 1 1 dx = ln |ax + b| ax + b a (1) (2) These restrictions are shown in the third column. \end{equation}, \begin{equation} 1. ∫ u n d u = u n + 1 n + 1 + C, n ≠ − 1 ∫ u n d u = u n + 1 n + 1 + C, n ≠ − 1. \int \sec x \ dx = \ln | \sec x + \tan x | = 2 \tanh^{-1} \left (\tan \frac{x}{2} \right) \int x^2 \ln x \ dx = \frac{1}{3} x^3 \ln x-\frac{x^3}{9} It includes: Table of Basic Forms; Table of Rational Integrals; Table of Integrals with Roots; Table of Integrals with Logarithms; Table of Exponential Integrals; Table of Trigonometric Integrals Forms … \int e^{ax^2}\ dx = -\frac{i\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(ix\sqrt{a}\right) Integration — is one of the main mathematical operations. \end{equation}, \begin{equation} \text{ where } \Gamma(a,x)=\int_x^{\infty} t^{a-1}e^{-t}\hspace{2pt}\text{d}t \int \ln ax\ dx = x \ln ax – x Administrator of Mini Physics. \int \tan^3 ax dx = \frac{1}{a} \ln \cos ax + \frac{1}{2a}\sec^2 ax 99. a \sin ax \sinh bx Students, teachers, parents, and everyone can find solutions to their math problems instantly. \frac{1}{a}\sqrt{ax^2+bx + c} \begin{array}{l} +\frac{1}{2}\left(x^2-\frac{b^2}{a^2}\right)\ln (ax+b) \end{equation}, \begin{equation} \\ \frac{2}{3} x(x-a)^{3/2} – \frac{4}{15} (x-a)^{5/2}, \text{ or} \end{equation}, \begin{equation} \int \csc^2 ax\ dx = -\frac{1}{a} \cot ax – \frac{\sin[(2a+b)x]}{4(2a+b)} \int \frac{x}{\sqrt{a^2-x^2}}\ dx = -\sqrt{a^2-x^2} + \frac{\sin bx}{2b} \end{equation}, \begin{equation} Use the table of integral formulas and the rules above to evaluate the following integrals. Table of Integrals Engineers usually refer to a table of integrals when performing calculations involving integration. \int x \sin^2 x \ dx = \frac{x^2}{4}-\frac{1}{8}\cos 2x – \frac{1}{4} x \sin 2x a \cos ax \sinh bx \int e^{-ax^2}\ dx = \frac{\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(x\sqrt{a}\right) \int\frac{dx}{(a^2+x^2)^{3/2}}=\frac{x}{a^2\sqrt{a^2+x^2}} Table of Standard Integrals 1. \end{equation}, \begin{equation} \int &x \sqrt{a x^2 + bx + c}\ dx = \frac{1}{48a^{5/2}}\left ( \int \sqrt{\frac{x}{a-x}}\ dx = -\sqrt{x(a-x)} \end{equation}, \begin{equation} \int e^{ax}\ dx = \frac{1}{a}e^{ax} 16.

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