# table of basic integrals

\int x e^x \cos x\ dx = \frac{1}{2}e^x (x \cos x \end{equation}, \begin{equation} \int \frac{\ln ax}{x}\ dx = \frac{1}{2}\left ( \ln ax \right)^2 \end{equation}, \begin{equation} Not to mention their servers 105. Table of Standard Integrals 1. \int x \sec^2 x \ dx = \ln \cos x + x \tan x +\frac{1}{2}\left(x^2-\frac{b^2}{a^2}\right)\ln (ax+b) \end{equation}, \begin{equation} \end{equation}, \begin{equation} \displaystyle{\frac{e^{2ax}}{4a} + \frac{x}{2}} & a = b \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:dewitt} \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 \end{array} \end{equation}, \begin{equation} \end{equation}, $$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2}} = \sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+ \infty}_{- \infty}x^{2n} e^{-ax^{2}} = (-1)^{n} \frac{\partial^{n}}{\partial a^{n}}\sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2} + bx} = e^{\frac{b^2}{4a}}\sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \sin^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 – \frac{6(-1)^n}{n^2 \pi^2} \right)$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \cos^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 + \frac{6(-1)^n}{n^2 \pi^2} \right)$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} xÂ \cos \left( \frac{ \pi x}{a} \right)Â \sin \left( \frac{2 \pi x}{a} \right) = \frac{8a^2}{9 \pi ^2}$$, $$\int\limits^{a}_{b} \frac{dx}{\sqrt{\left(a-x \right) \left(x-b \right)}} = \pi \text{ for a > b}$$, $$\int\limits^{a}_{b} \frac{dx}{x\sqrt{\left(a-x \right) \left(x-b \right)}} = \frac{ \pi}{\sqrt{ab}} \text{ for a > b > 0}$$, $$\int\limits^{\frac{\pi}{2}}_{- \frac{\pi}{2}} \frac{dx}{1+ y \sin x} = \frac{\pi}{\sqrt{1 – y^2}} \text{ for -1 < y < 1}$$, $$\int \frac{dx}{\sqrt{a^{2} – x^{2}}} = \text{arcsin} \, \frac{x}{a}$$, $$\int \frac{x dx}{\sqrt{a^{2} + x^{2}}} = \sqrt{a^{2} + x^{2}}$$, $$\int \frac{dx}{\sqrt{a^{2} +x^{2}}} = \text{ln} \, \left(x + \sqrt{a^{2} + x^{2}} \right)$$, $$\int \frac{dx}{a^{2} +x^{2}} = \frac{1}{a} \, \text{arctan} \, \frac{x}{a}$$, $$\int \frac{dx}{ \left( a^{2} + x^{2} \right)^{\frac{3}{2}}} = \frac{1}{a^{2}} \frac{x}{\sqrt{a^{2} +x^{2}}}$$, $$\int\frac{x \, dx}{ \left( a^{2}+x^{2} \right)^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{a^{2} + x^{2}}}$$, $$\int \frac{dx}{\sqrt{ (x – a)^{2} + b^{2}}} = \text{ln} \, \frac{1}{(a – x) + \sqrt{(a-x)^{2} + b^{2}}}$$, $$\int \frac{(x – a) \, dx}{\left[ (x-a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{(x-a)^{2} + b^{2}}}$$, $$\int \frac{dx}{\left[ (x – a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \frac{x – a}{b^{2} \sqrt{(x – a)^{2} +b^{2}}}$$. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \csc^3 x\ dx = -\frac{1}{2}\cot x \csc x + \frac{1}{2} \ln | \csc x – \cot x | \end{equation}, \begin{equation} \text{ where } \Gamma(a,x)=\int_x^{\infty} t^{a-1}e^{-t}\hspace{2pt}\text{d}t Table of Trig Integrals. -a\ln \left [ \sqrt{x} + \sqrt{x+a}\right] – \frac{\cos[(2a+b)x]}{4(2a+b)} \end{equation}, \begin{equation} $\int \frac{du}{u}=\text{ln}|u|+C$ 3. +\frac{i\sqrt{\pi}}{2a^{3/2}} \end{equation}, \begin{equation}\label{eq:ebke} \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 \end{equation}, \begin{equation} \int e^{ax}\ dx = \frac{1}{a}e^{ax} \end{equation}, \begin{equation} 108. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \sec^n x \tan x \ dx = \frac{1}{n} \sec^n x , n\ne 0 1. â« (1 / 2) ln (x) dx 2. â« [sin (x) + x 5] dx 3. â« [sinh (x) - 3] dx 4. â« - x sin (x) dx 5. -\frac{\sin[(2a-b)x]}{4(2a-b)} Basic forms. Free Integration Worksheet. \begin{cases} \frac{4ac-b^2}{8a^{3/2}}\ln \left| 2ax + b + 2\sqrt{a(ax^2+bx^+c)}\right | \int \cos ax \cosh bx\ dx = \displaystyle{ \frac{ e^{(a+2b)x}}{(a+2b)} \end{equation}, \begin{equation}\label{eq:swift2} You can verify any of the formulas by differentiating the function on the right side and obtaining the integrand. \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} – \int \cosh ax\ dx =\frac{1}{a} \sinh ax + \frac{\sin bx}{2b} 102. %PDF-1.5 \end{equation}, \begin{equation} \int \sin^n ax \ dx = \frac{1}{a^2 + b^2} \left[ \end{equation}, \begin{equation}\label{eq:qarles2} There have been visitors to integral-table.com since 2004. Notify me of follow-up comments by email. b \sin ax \sinh bx On this page, the tables contain examples of the most common integrals. /Filter /FlateDecode It is essential for mathematicians, scientists, and engineers, who rely on it when identifying and subsequently solving extremely complex problems. Integral tables >> Basic forms. \right] \end{equation}, \begin{equation}\label{eq:qarles1} \end{equation}, \begin{equation} 1, \frac{n+3}{2}, -\tan^2 ax \right) \int e^{bx} \sin ax\ dx = \frac{1}{a^2+b^2}e^{bx} (b\sin ax – a\cos ax) Integration is the basic operation in integral calculus. \int \tanh ax\hspace{1.5pt} dx =\frac{1}{a} \ln \cosh ax \end{equation}, \begin{equation} \int x^2 \sin x\ dx = \left(2-x^2\right) \cos x + 2 x \sin x \end{equation}, \begin{equation} \int x e^x\ dx = (x-1) e^x \int \cos ax \sin bx\ dx = \frac{\cos[(a-b) x]}{2(a-b)} – \int \sqrt{x} e^{ax}\ dx = \frac{1}{a}\sqrt{x}e^{ax} Forms containing logarithms and exponentials. – \frac{\sin[(2a+b)x]}{4(2a+b)} Integral Table. 16. \int \sec x \csc x \ dx = \ln | \tan x | \int \frac{x^2}{\sqrt{x^2 \pm a^2}}\ dx = \frac{1}{2}x\sqrt{x^2 \pm a^2} \frac{1}{2}, \frac{1-n}{2}, \frac{3}{2}, \cos^2 ax \int \sin^2 ax\ dx = \frac{x}{2} – \frac{\sin 2ax} {4a} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int &x \sqrt{a x^2 + bx + c}\ dx = \frac{1}{48a^{5/2}}\left ( \end{equation}, \begin{equation} Table of Basic Integrals Basic Forms 1 Z (1) xn dx = xn+1 , n 6= â1 n+1 1 Z (2) dx = ln |x| x Z Z (3) udv = uv â vdu 1 1 Z (4) dx = ln |ax + b| ax + b a Integrals of Rational Functions 1 1 Z (5) 2 dx = â (x + a) x+a (x + a)n+1 Z n (6) (x + a) dx = , n 6= â1 n+1 (x + a)n+1 ((n + 1)x â a) Z (7) x(x + a)n dx = (n + 1)(n + 2) 1 Z (8) dx = tanâ1 x 1 + x2 1 1 Z â1 x (9) dx = tan a2 + x2 a a 1 1 Z x (10) dx = ln |a2 + x2 | a2 +x 2 2 \frac{\sin[2(a-b)x]}{16(a-b)} \end{equation}, \begin{equation}\label{eq:Winokur2} – \frac{\cos bx}{2b} \int \cos ax \sinh bx\ dx = 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. \int x \sqrt{ax + b}\ dx = Types of Integrals. 4)>$�ÿ�K��1��~)���$��z!~Z��dBPb�H2͈к$��*��'�z�E���D�S#J���t�u�aլM��$.1�����8Q���q3Ds�d-���YOeU)(h��$�Dp�XBm� \int \sinh ax \cosh ax dx= \int x \cos x \ dx = \cos x + x \sin x \\ \frac{2}{15}(2a+3x)(x-a)^{3/2} -\frac{1}{a}{\cos ax} \hspace{2mm}{_2F_1}\left[ \int x \sin ax\ dx = -\frac{x \cos ax}{a} + \frac{\sin ax}{a^2} \int \frac{x}{\sqrt{ax^2+bx+c}}\ dx= \int \ln ( x^2 + a^2 )\hspace{.5ex} {dx} = x \ln (x^2 + a^2 ) +2a\tan^{-1} \frac{x}{a} – 2x Example: 13. (A) The Power Rule : Examples : d dx {un} = nu nâ1. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} Home University Mathematics Integration Table, \begin{equation} \int \frac{1}{(x+a)^2}dx = -\frac{1}{x+a} Tinycards by Duolingo is a fun flashcard app that helps you memorize anything for free, forever. Table of Integrals Engineers usually refer to a table of integrals when performing calculations involving integration. \end{equation}, \begin{equation} %���� 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 The table presents a selection of integrals found in the Calculus books. \int \sin^2 ax \cos bx\ dx = \int \cos^p ax dx = -\frac{1}{a(1+p)}{\cos^{1+p} ax} \times 2 \sqrt{a} \sqrt{ax^2+bx+c} \int \frac{x}{ax^2+bx+c}dx = \frac{1}{2a}\ln|ax^2+bx+c| \int (\ln x)^2\ dx = 2x – 2x \ln x + x (\ln x)^2 \end{equation}, \begin{equation} \int \frac{x}{\sqrt{a^2-x^2}}\ dx = -\sqrt{a^2-x^2} 4. \end{equation}, \begin{equation} \int x^n \cos ax \ dx = 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} 103. Forms containing inverse trigonometric functions. \int \sin^2 ax \cos^2 bx dx = \frac{x}{4} 7. \frac{b+2ax}{4a}\sqrt{ax^2+bx+c} b \cos ax \cosh bx + \displaystyle{\frac{e^{ax}-2\tan^{-1}[e^{ax}]}{a} } & a = b All the immediate integrals. 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. \right . = uv " vdu ! \int \frac{1}{(x+a)(x+b)}dx = \frac{1}{b-a}\ln\frac{a+x}{b+x}, \text{ } a\ne b {_2F_1}\left[ 1+\frac{a}{2b},1,2+\frac{a}{2b}, -e^{2bx}\right] }& \\ Administrator of Mini Physics. (x+a)ndx=(x+a)n a 1+n + x 1+n " #$ % &', n! \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 \int \sin ax \ dx = -\frac{1}{a} \cos ax \frac{b}{12a}- \end{cases} Not to mention their servers gave up the ghost turned into Zombies on 25 March 2015 (Brains! \int x \ln (ax + b)\ dx = \frac{bx}{2a}-\frac{1}{4}x^2 101. \int\sqrt{x^2 \pm a^2}\ dx = \frac{1}{2}x\sqrt{x^2\pm a^2} \end{equation}, \begin{equation} \frac{1}{2}\left( x^2 – \frac{a^2}{b^2} \right ) \ln \left (a^2 -b^2 x^2 \right) 98. \end{equation}, \begin{equation} \int \csc^nx \cot x\ dx = -\frac{1}{n}\csc^n x, n\ne 0 Free math lessons and math homework help from basic math to algebra, geometry and beyond. \end{equation}, \begin{equation} \int (x+a)^n dx = \frac{(x+a)^{n+1}}{n+1}, n\ne -1 10. 31. \int \sqrt{a^2 – x^2}\ dx = \frac{1}{2} x \sqrt{a^2-x^2} stream -2ax + \sinh 2ax \right] \right] \int x \sin^2 x \ dx = \frac{x^2}{4}-\frac{1}{8}\cos 2x – \frac{1}{4} x \sin 2x \int x e^x \sin x\ dx = \frac{1}{2}e^x (\cos x – x \cos x + x \sin x) 111. Table of Integralsâ. \int \frac{1}{\sqrt{x\pm a}}\ dx = 2\sqrt{x\pm a} 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} 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 \end{equation}, \begin{equation}\label{eq:Russ} Table of Indefinite Integral Formulas . \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \ln (ax + b) \ dx = \left ( x + \frac{b}{a} \right) \ln (ax+b) – x , a\ne 0 \end{equation}, \begin{equation} This leaï¬et provides such a table. \int e^{ax} \sinh bx \ dx = \frac{2}{15 a^2}(-2b^2+abx + 3 a^2 x^2) \frac{1}{4a}\left[ [Note that you may need to use more than one of the above rules for one integral]. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \frac{\cos[(a+b)x]}{2(a+b)} , a\ne b \begin{array}{l} \frac{1}{a^2 + b^2} \left[ For the following, the letters a, b, n, and C represent constants.. \end{equation}, \begin{equation}\label{eq:Winokur1} \int \tan ax\ dx = -\frac{1}{a} \ln \cos ax \end{equation}, \begin{equation} 99. \pm\frac{1}{2}a^2 \ln \left | x + \sqrt{x^2\pm a^2} \right | \end{equation}, \begin{equation} Basic Integrals; Trigonometric Integrals; Exponential and Logarithmic Integrals; Hyperbolic Integrals; Inverse Trigonometric Integrals; Integrals Involving a2 + u2, a > 0; Integrals Involving u2 â a2, a > 0; Integrals Involving a2 â u2, a > 0; Integrals Involving 2au â u2, a > 0; Integrals â¦ \int x^n \cos x dx = \int e^{ax^2}\ dx = -\frac{i\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(ix\sqrt{a}\right) (x)dx RATIONAL FUNCTIONS (5) 1 ax+b!dx= 1 a ln(ax+b) (6) 1 (x+a)2!dx= "1 x+a (7)! \int \frac{x}{\sqrt{x^2\pm a^2}}\ dx = \sqrt{x^2 \pm a^2} \int \cos ax\ dx= \frac{1}{a} \sin ax >> \end{equation}, \begin{equation}\label{eq:ajoy} \frac{1}{a^2 + b^2} \left[ \int x^n \sin x \ dx = -\frac{1}{2}(i)^n\left[ \Gamma(n+1, -ix) \int \ln ax\ dx = x \ln ax – x \mp \frac{1}{2}a^2 \ln \left| x + \sqrt{x^2\pm a^2} \right | \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*��>\�+�. \int \frac{1}{ax+b}dx = \frac{1}{a} \ln |ax + b| \end{equation}, \begin{equation}\label{eq:veky} \int \sqrt{\frac{x}{a+x}}\ dx = \sqrt{x(a+x)} \left( – 3b^2 + 2 abx + 8 a(c+ax^2) \right) \end{equation}, \begin{equation} \int x^n e^{ax}\ dx = \dfrac{x^n e^{ax}}{a} – \end{equation}, \begin{equation} \end{equation}, \begin{equation} This page lists some of the most common antiderivatives. \int (ax+b)^{3/2}\ dx =\frac{2}{5a}(ax+b)^{5/2} a \cos ax \sinh bx \end{equation}, \begin{equation} Basic Integrals. 113. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{cases} Forms â¦ \int u dv = uv – \int v du 100. \dfrac{n}{a}\int x^{n-1}e^{ax}\hspace{1pt}\text{d}x These restrictions are shown in the third column. \int \sin ax \sinh bx \ dx = \int \frac{x}{a^2+x^2}dx = \frac{1}{2}\ln|a^2+x^2| The clustrmap is periodically (and automatically) archived and its counters reset, so the total is smaller. \frac{1}{a}\sqrt{ax^2+bx + c} \end{equation}, \begin{equation} 110. +\frac{1}{2}a^2\tan^{-1}\frac{x}{\sqrt{a^2-x^2}} \int x \cos ax \ dx = \frac{1}{a^2} \cos ax + \frac{x}{a} \sin ax \end{equation}, \begin{equation} \end{equation}, \begin{equation} ÑÐ°Ð±Ð»Ð¸ÑÐ° Ð¸Ð½ÑÐµÐ³ÑÐ°Ð»Ð¾Ð². \int \sin^2 x \cos x\ dx = \frac{1}{3} \sin^3 x Students, teachers, parents, and everyone can find solutions to their math problems instantly. \end{equation}, \begin{equation} \frac{1}{b^2-a^2}\left[ Use the table of integral formulas and the rules above to evaluate the following integrals. \int x^n dx = \frac{1}{n+1}x^{n+1},\hspace{1ex}n\neq -1 \\ & \left. \sqrt{x^3(ax+b)} + -\frac{\sin 2ax}{8a}- \end{equation}, \begin{equation} 107. \end{equation}, \begin{equation}\label{eq:Larry-Morris}\begin{split} \end{equation}, \begin{equation} \displaystyle{\frac{e^{2ax}}{4a} – \frac{x}{2}} & a = b + 3(b^3-4abc)\ln \left|b + 2ax + 2\sqrt{a}\sqrt{ax^2+bx+c} \right| \right) \begin{cases} \frac{1+p}{2}, \frac{1}{2}, \frac{3+p}{2}, \cos^2 ax Read Free Table Of Integrals Integral Table periodically (and automatically) archived and its counters reset, so the total is smaller. 28. \end{equation}, \begin{equation} -a \cos ax \cosh bx + \end{equation}, \begin{equation} \frac{b^3}{8a^{5/2}}\ln \left | a\sqrt{x} + \sqrt{a(ax+b)} \right | 104. \end{equation}, \begin{equation} \int \sinh ax\ dx = \frac{1}{a} \cosh ax \end{equation}, \begin{equation}\label{eq:Kloeppel} + (-1)^n \Gamma(n+1, ix)\right] \int \sec^2 x \tan x\ dx = \frac{1}{2} \sec^2 x \end{equation}, \begin{equation} The clustrmap is Page 13/24. \end{equation}, \begin{equation} \frac{1}{\sqrt{a}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right | \end{equation}, \begin{equation} \int x \ln x \ dx = \frac{1}{2} x^2 \ln x-\frac{x^2}{4} \int \frac{1}{ax^2+bx+c}dx = \frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} -b^2 \ln \left| a\sqrt{x} + \sqrt{a(ax+b)} \right| \right ] \end{equation}, \begin{equation} Basic Differentiation Rules Basic Integration Formulas DERIVATIVES AND INTEGRALS © Houghton Mifflin Company, Inc. 1. \int (\ln x)^3\ dx = -6 x+x (\ln x)^3-3 x (\ln x)^2+6 x \ln x \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int x \ln \left ( a^2 – b^2 x^2 \right )\ dx = -\frac{1}{2}x^2+ \int \frac{\ln x}{x^2}\ dx = -\frac{1}{x}-\frac{\ln x}{x} {_2}F_1\left( \frac{n+1}{2}, \int \cos^2 ax\ dx = \frac{x}{2}+\frac{ \sin 2ax}{4a} \end{equation}, \begin{equation} \int \frac{1}{\sqrt{x^2 \pm a^2}}\ dx = \ln \left | x + \sqrt{x^2 \pm a^2} \right | ��H�$e���׍� �XH*N�"���뷿�u7M>$4��������kffgJ&��N9�N'�jB�Mn�ۅ����C�ȄQ��}����n�*��Y�����a����� � \right] \end{equation}, \begin{equation} Apr 30, 2018 - Complete table of integrals in a single sheet. = 1 n + 1 x n + 1 (2) 1 x dx ! +\frac{\sin 2bx}{8b}- \end{equation}, \begin{equation}\label{eq:Duley} \end{equation}, \begin{equation}\label{eq:ritzert} \int \csc x\ dx = \ln \left | \tan \frac{x}{2} \right| = \ln | \csc x – \cot x| + C \end{equation}, \begin{equation} \int \sin^2 ax \cos^2 ax\ dx = \frac{x}{8}-\frac{\sin 4ax}{32a} \int \frac{1}{\sqrt{a-x}}\ dx = -2\sqrt{a-x} \int \sec^2 ax\ dx = \frac{1}{a} \tan ax \int \frac{x}{(x+a)^2}dx = \frac{a}{a+x}+\ln |a+x| \end{equation}, \begin{equation} \int \cos^2 ax \sin ax\ dx = -\frac{1}{3a}\cos^3{ax} \text{erf}\left(i\sqrt{ax}\right), \end{equation}, \begin{equation} \int x(x+a)^n dx = \frac{(x+a)^{n+1} ( (n+1)x-a)}{(n+1)(n+2)} } \end{equation}, \begin{equation} – \sin x + x \sin x) \int \tan^n ax\ dx = \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ a \cosh bx – b \sinh bx ] & a\ne b \\ \end{equation}, \begin{equation} a \sin ax \sinh bx \int x^n e^{ax}\ dx = \frac{(-1)^n}{a^{n+1}}\Gamma[1+n,-ax], 112. \int \tan^3 ax dx = \frac{1}{a} \ln \cos ax + \frac{1}{2a}\sec^2 ax \end{cases} \int \cos^2 ax \sin bx\ dx = \frac{\cos[(2a-b)x]}{4(2a-b)} b \cosh bx \sinh ax Integration â is one of the main mathematical operations. \right] Table of Integrals. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \tan^2 ax\ dx = -x + \frac{1}{a} \tan ax \int \frac{1}{\sqrt{a^2 – x^2}}\ dx = \sin^{-1}\frac{x}{a} \text{ where erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt \right. & a\ne b \\ 7 0 obj << It is a compilation of the most commonly used integrals. \end{equation}, \begin{equation}\label{eq:Rigo} \left\{ \int \sinh ax \cosh bx \ dx = \int \sec x \ dx = \ln | \sec x + \tan x | = 2 \tanh^{-1} \left (\tan \frac{x}{2} \right) \end{equation}, \begin{equation}\label{eq:swift1} {_2F_1}\left[ \int \sqrt{a x^2 + b x + c}\ dx = 19. \int \ln ( x^2 – a^2 )\hspace{.5ex} {dx} = x \ln (x^2 – a^2 ) +a\ln \frac{x+a}{x-a} – 2x \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ -b \cosh bx + a \sinh bx ] & a\ne b \\ \displaystyle{ -a\tan^{-1}\frac{\sqrt{x(a-x)}}{x-a} \end{equation}, \begin{equation} 34. \int \sqrt{x(ax+b)}\ dx = \frac{1}{4a^{3/2}}\left[(2ax + b)\sqrt{ax(ax+b)} Table of Integrals BASIC FORMS (1) x n dx ! \int \frac{1}{1+x^2}dx = \tan^{-1}x \int e^{bx} \cos ax\ dx = \frac{1}{a^2 + b^2} e^{bx} ( a \sin ax + b \cos ax ) \end{equation}, \begin{equation} \frac{\tan^{n+1} ax }{a(1+n)} \times \end{equation}, \begin{equation} \end{equation}, \begin{equation} Indefinite integrals. -\Gamma(n+1, ixa)\right] \right] Sometimes restrictions need to be placed on the values of some of the variables. \int \frac{x^2}{a^2+x^2}dx = x-a\tan^{-1}\frac{x}{a} \int x^2 \cos ax \ dx = \frac{2 x \cos ax }{a^2} + \frac{ a^2 x^2 – 2 }{a^3} \sin ax \int \frac{x}{\sqrt{x\pm a} } \ dx = \frac{2}{3}(x\mp 2a)\sqrt{x\pm a} \int x^2 e^{ax}\ dx = \left(\frac{x^2}{a}-\frac{2x}{a^2}+\frac{2}{a^3}\right) e^{ax} \int \frac{x^3}{a^2+x^2}dx = \frac{1}{2}x^2-\frac{1}{2}a^2\ln|a^2+x^2| Table of integrals - the basic formulas of indefinite integrals. \int \sqrt{ax+b}\ dx = \left(\frac{2b}{3a}+\frac{2x}{3}\right)\sqrt{ax+b} Tables contain examples of the formulas by differentiating the function on the values of some of the following entries. 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