# table of basic integrals

\frac{b^2}{8a^2x}+ \int e^{ax} \sinh bx \ dx = \begin{cases} \begin{cases} \int \sqrt{x(ax+b)}\ dx = \frac{1}{4a^{3/2}}\left[(2ax + b)\sqrt{ax(ax+b)} \end{equation}, \begin{equation} \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}{\sqrt{ax^2+bx+c}}\ dx= Notify me of follow-up comments by email. Integral tables >> Basic forms. ÑÐ°Ð±Ð»Ð¸ÑÐ° Ð¸Ð½ÑÐµÐ³ÑÐ°Ð»Ð¾Ð². \int \cos^3 ax dx = \frac{3 \sin ax}{4a}+\frac{ \sin 3ax}{12a} 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 \end{equation}, \begin{equation} \int \tan ax\ dx = -\frac{1}{a} \ln \cos ax 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) \end{equation}, \begin{equation}\label{eq:swift2} \int x^2 \ln x \ dx = \frac{1}{3} x^3 \ln x-\frac{x^3}{9} Free Integration Worksheet. \int x \cos^2 x \ dx = \frac{x^2}{4}+\frac{1}{8}\cos 2x + \frac{1}{4} x \sin 2x +\frac{1}{2}\left(x^2-\frac{b^2}{a^2}\right)\ln (ax+b) \int \ln ( x^2 + a^2 )\hspace{.5ex} {dx} = x \ln (x^2 + a^2 ) +2a\tan^{-1} \frac{x}{a} – 2x \end{equation}, \begin{equation} \int \frac{1}{\sqrt{a-x}}\ dx = -2\sqrt{a-x} \int x^n e^{ax}\ dx = \dfrac{x^n e^{ax}}{a} – – a \cosh ax \sinh bx \right] \int \frac{1}{x}dx = \ln |x| \end{equation}, \begin{equation} %���� \hspace{1cm}-\frac{1}{a}e^{ax}{_2F_1}\left[ 1, \frac{a}{2b},1+\frac{a}{2b}, -e^{2bx}\right] \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:Russ} These tinycards help you memorize the table of basic integrals. \int \sinh ax \cosh bx \ dx = \int \sqrt{ax+b}\ dx = \left(\frac{2b}{3a}+\frac{2x}{3}\right)\sqrt{ax+b} \int \cos ax \cosh bx\ dx = \int (ax+b)^{3/2}\ dx =\frac{2}{5a}(ax+b)^{5/2} (A) The Power Rule : Examples : d dx {un} = nu nâ1. \int \cos^2 ax \sin ax\ dx = -\frac{1}{3a}\cos^3{ax} b \cos ax \cosh bx + \frac{1}{\sqrt{a}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right | \end{equation}, \begin{equation} \int \csc x\ dx = \ln \left | \tan \frac{x}{2} \right| = \ln | \csc x – \cot x| + C \end{equation}, \begin{equation} 1. \end{equation}, \begin{equation} + \frac{\sin bx}{2b} \int \cos ax\ dx= \frac{1}{a} \sin ax (3x2 + 4)d dx {u} = 12 u.u d dx { 2 â 4x2 + 7x5} = 1 2 2 â 4x2 + 7x5 (â8x + 35x4) d dx {c} = 0 , c is a constant ddx {6} = 0 , since â 3.14 is a constant. \frac{1}{a^2 + b^2} \left[ \end{equation}, \begin{equation} \end{cases} \end{equation}, \begin{equation} -\frac{\sin 2ax}{8a}- \int x^2 \cos ax \ dx = \frac{2 x \cos ax }{a^2} + \frac{ a^2 x^2 – 2 }{a^3} \sin ax \end{equation}, \begin{equation} 101. \sqrt{x^3(ax+b)} + All the immediate integrals. b \cosh bx \sin ax – \end{equation}, \begin{equation} \int \sec x \tan x\ dx = \sec x {_2F_1}\left[ 1+\frac{a}{2b},1,2+\frac{a}{2b}, -e^{2bx}\right] }& \\ ��H�$e���׍� �XH*N�"���뷿�u7M>$4��������kffgJ&��N9�N'�jB�Mn�ۅ����C�ȄQ��}����n�*��Y�����a����� � \end{equation}, \begin{equation} stream \end{equation}, \begin{equation} \int x \ln x \ dx = \frac{1}{2} x^2 \ln x-\frac{x^2}{4} \end{equation}, \begin{equation} Read Free Table Of Integrals Integral Table periodically (and automatically) archived and its counters reset, so the total is smaller. \int \sec^3 x \ {dx} = \frac{1}{2} \sec x \tan x + \frac{1}{2}\ln | \sec x + \tan x | 4)>$�ÿ�K��1��~)���$��z!~Z��dBPb�H2͈к$��*��'�z�E���D�S#J���t�u�aլM��$.1�����8Q���q3Ds�d-���YOeU)(h��$�Dp�XBm� If you spot any errors or want to suggest improvements, please contact us. \end{equation}, \begin{equation} \frac{b}{2a^{3/2}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right | 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. \int e^{ax^2}\ dx = -\frac{i\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(ix\sqrt{a}\right) Use the table of integral formulas and the rules above to evaluate the following integrals. \end{equation}, \begin{equation} \int e^{ax} \tanh bx\ dx = \end{equation}, \begin{equation} \int x e^x \sin x\ dx = \frac{1}{2}e^x (\cos x – x \cos x + x \sin x) 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! \int \ln (ax + b) \ dx = \left ( x + \frac{b}{a} \right) \ln (ax+b) – x , a\ne 0 \end{equation}, \begin{equation} Z xndx= xn+1 n+1 +C (n6= 1) 2. [Note that you may need to use more than one of the above rules for one integral]. Apr 30, 2018 - Complete table of integrals in a single sheet. \end{equation}, \begin{equation} \frac{1+p}{2}, \frac{1}{2}, \frac{3+p}{2}, \cos^2 ax \int e^{ax} \cosh bx \ dx = Table of Standard Integrals 1. \displaystyle{\frac{e^{ax}-2\tan^{-1}[e^{ax}]}{a} } & a = b \end{equation}, \begin{equation} \frac{1}{a^2 + b^2} \left[ \int x^2 e^{x}\ dx = \left(x^2 – 2x + 2\right) e^{x} \int \sin ax \sinh bx \ dx = \end{equation}, \begin{equation}\label{eq:dewitt} \int \frac{x}{a^2+x^2}dx = \frac{1}{2}\ln|a^2+x^2| \int \cos^2 ax \sin bx\ dx = \frac{\cos[(2a-b)x]}{4(2a-b)} \int x^n \sin x \ dx = -\frac{1}{2}(i)^n\left[ \Gamma(n+1, -ix) \end{equation}, \begin{equation} \int x \ln \left ( a^2 – b^2 x^2 \right )\ dx = -\frac{1}{2}x^2+ \end{equation}, \begin{equation}\label{eq:swift1} Basic Integrals. x��ZIs�F��W�V�v�KR9$�qj\SS5c�e�h \frac{x}{3}\right] \end{equation}, \begin{equation}\label{eq:qarles1} \right] \int \sqrt{a x^2 + b x + c}\ dx = \int x^n e^{ax}\ dx = \frac{(-1)^n}{a^{n+1}}\Gamma[1+n,-ax], It is a compilation of the most commonly used integrals. \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} \frac{\cos[(a+b)x]}{2(a+b)} , a\ne b 1. \int e^x \sin x \ dx = \frac{1}{2}e^x (\sin x – \cos x) 28. \int x \sqrt{x^2 \pm a^2}\ dx= \frac{1}{3}\left ( x^2 \pm a^2 \right)^{3/2} The table presents a selection of integrals found in the Calculus books. 7. It is essential for mathematicians, scientists, and engineers, who rely on it when identifying and subsequently solving extremely complex problems. \end{equation}, \begin{equation} 31. \right] \frac{b^3}{8a^{5/2}}\ln \left | a\sqrt{x} + \sqrt{a(ax+b)} \right | \end{equation}, \begin{equation} Scroll down the page if you need more examples and step by step solutions of indefinite integrals. \int \sin^2 x \cos x\ dx = \frac{1}{3} \sin^3 x \int \tan^2 ax\ dx = -x + \frac{1}{a} \tan ax -2ax + \sinh 2ax \right] \int x e^{ax}\ dx = \left(\frac{x}{a}-\frac{1}{a^2}\right) e^{ax} \int \frac{x}{\sqrt{a^2-x^2}}\ dx = -\sqrt{a^2-x^2} \end{equation}, \begin{equation} On this page, the tables contain examples of the most common integrals. \mp \frac{1}{2}a^2 \ln \left| x + \sqrt{x^2\pm a^2} \right | \int \frac{1}{1+x^2}dx = \tan^{-1}x \end{equation}, \begin{equation} \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ -b \cosh bx + a \sinh bx ] & a\ne b \\ \end{equation}, \begin{equation} \int e^{-ax^2}\ dx = \frac{\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(x\sqrt{a}\right) /Length 2403 Free Table of Integrals to print on a single sheet side and side. \end{equation}, \begin{equation} = ln x (3) udv ! \end{equation}, \begin{equation}\label{eq:Larry-Morris}\begin{split} \int x e^x\ dx = (x-1) e^x 34. \int x^n \cos x dx = \end{equation}, \begin{equation} {_2F_1}\left[ \end{equation}, \begin{equation} \int \frac{x}{\sqrt{x\pm a} } \ dx = \frac{2}{3}(x\mp 2a)\sqrt{x\pm a} \int \sqrt{\frac{x}{a-x}}\ dx = -\sqrt{x(a-x)} \int \sin^2 ax\ dx = \frac{x}{2} – \frac{\sin 2ax} {4a} \end{equation}, \begin{equation}\label{eq:ritzert} \int \sqrt{a^2 – x^2}\ dx = \frac{1}{2} x \sqrt{a^2-x^2} Forms containing trigonometric functions. \int x^2 \sin ax\ dx =\frac{2-a^2x^2}{a^3}\cos ax +\frac{ 2 x \sin ax}{a^2} \int \sqrt{x-a}\ dx = \frac{2}{3}(x-a)^{3/2} $\int {u}^{n}du=\frac{{u}^{n+1}}{n+1}+C,n\ne \text{â}1$ 2. \end{equation}, \begin{equation}\label{eq:swift3} \end{equation}, \begin{equation} \int x^n dx = \frac{1}{n+1}x^{n+1},\hspace{1ex}n\neq -1 This leaï¬et provides such a table. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \frac{1}{a}\sqrt{ax^2+bx + c} b \cosh bx \sinh ax \int \cos ax \sin bx\ dx = \frac{\cos[(a-b) x]}{2(a-b)} – \int\sqrt{x^2 \pm a^2}\ dx = \frac{1}{2}x\sqrt{x^2\pm a^2} \int u dv = uv – \int v du 108. \left\{ \end{equation}, \begin{equation} \frac{1}{4a}\left[ \int x(x+a)^n dx = \frac{(x+a)^{n+1} ( (n+1)x-a)}{(n+1)(n+2)} \end{equation}, \begin{equation} \int \sec x \csc x \ dx = \ln | \tan x | +\frac{i\sqrt{\pi}}{2a^{3/2}} \int x e^{-ax^2}\ {dx} = -\dfrac{1}{2a}e^{-ax^2} – (-1)^n\Gamma(n+1, -ix)\right] \int \cos^p ax dx = -\frac{1}{a(1+p)}{\cos^{1+p} ax} \times \end{equation}, \begin{equation} 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. \right] Table of integrals - the basic formulas of indefinite integrals. \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 2 \sqrt{a} \sqrt{ax^2+bx+c} Home University Mathematics Integration Table, \begin{equation} \end{equation}, \begin{equation}\label{eq:xul} 99. \int \sin^2 ax \cos^2 bx dx = \frac{x}{4} \int e^{bx} \cos ax\ dx = \frac{1}{a^2 + b^2} e^{bx} ( a \sin ax + b \cos ax ) \int \sqrt{x^3(ax+b)} \ dx =\left [ \pm\frac{1}{2}a^2 \ln \left | x + \sqrt{x^2\pm a^2} \right | Integrals Involving a + bu, a â  0. 110. 102. Table of Trig Integrals. \end{equation}, \begin{equation} \int (\ln x)^2\ dx = 2x – 2x \ln x + x (\ln x)^2 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¦ \end{equation}, \begin{equation} \int e^{bx} \sin ax\ dx = \frac{1}{a^2+b^2}e^{bx} (b\sin ax – a\cos ax) b \sin ax \sinh bx \int \frac{1}{ax^2+bx+c}dx = \frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} \end{equation}, \begin{equation} -a\ln \left [ \sqrt{x} + \sqrt{x+a}\right] Forms containing logarithms and exponentials. 22. \end{equation}, \begin{equation} Basic Differentiation Rules Basic Integration Formulas DERIVATIVES AND INTEGRALS © Houghton Mifflin Company, Inc. 1. \right] 13. \int \frac{x^2}{\sqrt{x^2 \pm a^2}}\ dx = \frac{1}{2}x\sqrt{x^2 \pm a^2} Basic Integrals. \frac{1}{2}, \frac{1-n}{2}, \frac{3}{2}, \cos^2 ax Integral Table. +\frac{\sin 2bx}{8b}- \int e^{ax}\ dx = \frac{1}{a}e^{ax} \int \csc^nx \cot x\ dx = -\frac{1}{n}\csc^n x, n\ne 0 \end{equation}, \begin{equation} \int \sinh ax \cosh ax dx= The clustrmap is periodically (and automatically) archived and its counters reset, so the total is smaller. \int x \sec^2 x \ dx = \ln \cos x + x \tan x \end{equation}, \begin{equation} \right . \end{equation}, \begin{equation} Table of Integralsâ. 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 \end{equation}, \begin{equation} $\int \frac{du}{u}=\text{ln}|u|+C$ 3. \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:Gilmore} \int \tan^n ax\ dx = – \end{equation}, \begin{equation} \frac{1}{a^2 + b^2} \left[ \end{equation}, \begin{equation} 4. Not to mention their servers \int &x \sqrt{a x^2 + bx + c}\ dx = \frac{1}{48a^{5/2}}\left ( – \sin x + x \sin x) -\Gamma(n+1, ixa)\right] 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. \end{equation}, \begin{equation} 98. ���_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*��>\�+�. 1, \frac{n+3}{2}, -\tan^2 ax \right) -2x + \left( \frac{b}{2a}+x \right )\ln \left (ax^2+bx+c \right) \int x \cos ax \ dx = \frac{1}{a^2} \cos ax + \frac{x}{a} \sin ax \end{equation}, \begin{equation} Table of Integrals Engineers usually refer to a table of integrals when performing calculations involving integration. \int \frac{1}{(x+a)^2}dx = -\frac{1}{x+a} Basic forms. \end{equation}, \begin{equation} \int x \ln (ax + b)\ dx = \frac{bx}{2a}-\frac{1}{4}x^2 \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:veky} \end{equation}, \begin{equation} \int \sinh ax\ dx = \frac{1}{a} \cosh ax \int \sin^3 ax \ dx = -\frac{3 \cos ax}{4a} + \frac{\cos 3ax} {12a} \int \frac{x}{(x+a)^2}dx = \frac{a}{a+x}+\ln |a+x| \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int x \sin^2 x \ dx = \frac{x^2}{4}-\frac{1}{8}\cos 2x – \frac{1}{4} x \sin 2x \end{cases} \int \frac{x}{\sqrt{x^2\pm a^2}}\ dx = \sqrt{x^2 \pm a^2} \end{equation}, \begin{equation} \int x^3 e^{x}\ dx = \left(x^3-3x^2 + 6x – 6\right) e^{x} \end{equation}, \begin{equation} \int \sqrt{x} e^{ax}\ dx = \frac{1}{a}\sqrt{x}e^{ax} \end{equation}, \begin{equation} 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. a \sin ax \cosh bx + b \cos ax \sinh bx \int \cos ax \sinh bx\ dx = \end{equation}, \begin{equation} \displaystyle{ \frac{ e^{(a+2b)x}}{(a+2b)} \int x^2 \cos x \ dx = 2 x \cos x + \left ( x^2 – 2 \right ) \sin x \displaystyle{\frac{e^{2ax}}{4a} – \frac{x}{2}} & a = b \int \ln ( x^2 – a^2 )\hspace{.5ex} {dx} = x \ln (x^2 – a^2 ) +a\ln \frac{x+a}{x-a} – 2x \end{equation}, \begin{equation}\label{eq:Rigo} A: TABLE OF BASIC DERIVATIVES Let u = u(x) be a differentiable function of the independent variable x, that is u(x) exists. \int \tanh ax\hspace{1.5pt} dx =\frac{1}{a} \ln \cosh ax Indefinite integrals. \int \tan^3 ax dx = \frac{1}{a} \ln \cos ax + \frac{1}{2a}\sec^2 ax \end{equation}, \begin{equation} \end{equation}, \begin{equation} a \sin ax \sinh bx Free math lessons and math homework help from basic math to algebra, geometry and beyond. \end{equation}, \begin{equation} You can verify any of the formulas by differentiating the function on the right side and obtaining the integrand. \end{array} \frac{1}{2}\left( x^2 – \frac{a^2}{b^2} \right ) \ln \left (a^2 -b^2 x^2 \right) + \end{equation}, \begin{equation} \end{equation}, \begin{equation} 111. \end{equation}, \begin{equation} \int \sin^2 ax \cos bx\ dx = \frac{1}{2}(ia)^{1-n}\left [ (-1)^n \Gamma(n+1, -iax) /Filter /FlateDecode & a\ne b \\ \int \frac{1}{\sqrt{x\pm a}}\ dx = 2\sqrt{x\pm a} \end{equation}, \begin{equation} Integration is the basic operation in integral calculus. \end{equation}, \begin{equation} %PDF-1.5 \end{equation}, \begin{equation}\label{eq:qarles2} \end{equation}, \begin{equation} \int \sin^n ax \ dx = \end{equation}, \begin{equation} This page lists some of the most common antiderivatives. \end{equation}, \begin{equation}\label{eq:Winokur1} \end{equation}, \begin{equation} \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}} 113. (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} \int \frac{1}{ax+b}dx = \frac{1}{a} \ln |ax + b| \frac{\tan^{n+1} ax }{a(1+n)} \times \end{equation}, \begin{equation} \displaystyle{\frac{e^{2ax}}{4a} + \frac{x}{2}} & a = b Table of Integrals. \int \frac{1}{a^2+x^2}dx = \frac{1}{a}\tan^{-1}\frac{x}{a} \end{equation}, \begin{equation} \int \frac{\ln x}{x^2}\ dx = -\frac{1}{x}-\frac{\ln x}{x} \int \sin ax \ dx = -\frac{1}{a} \cos ax 10. Integration â is one of the main mathematical operations. Forms containing inverse trigonometric functions. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:ebke} \end{equation}, \begin{equation} \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 \displaystyle{ + (-1)^n \Gamma(n+1, ix)\right] u ddx {(x3 + 4x + 1)3/4} = 34 (x3 + 4x + 1)â1/4. \end{equation}, \begin{equation} \\ \frac{2}{15}(2a+3x)(x-a)^{3/2} \end{equation}, \begin{equation} \end{equation}, \begin{equation} 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. a \cos ax \sinh bx -\frac{\sin[(2a-b)x]}{4(2a-b)} \end{equation}, \begin{equation} >> \end{equation}, \begin{equation} For the following, the letters a, b, n, and C represent constants.. \int \frac{1}{\sqrt{x^2 \pm a^2}}\ dx = \ln \left | x + \sqrt{x^2 \pm a^2} \right | \end{equation}, \begin{equation}\label{eq:Duley} \frac{b}{12a}- \end{equation}, \begin{equation} -a\tan^{-1}\frac{\sqrt{x(a-x)}}{x-a} \end{equation}, \begin{equation} Note: Most of the following integral entries are written for indefinite integrals, but they also apply to definite integrals. \int \cos^2 ax\ dx = \frac{x}{2}+\frac{ \sin 2ax}{4a} 112. \end{split} \\ & \left. 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 } 7 0 obj << Tinycards by Duolingo is a fun flashcard app that helps you memorize anything for free, forever. \end{equation}, \begin{equation} \int \sin^2 ax \cos^2 ax\ dx = \frac{x}{8}-\frac{\sin 4ax}{32a} 107. \end{equation}, \begin{equation} – \frac{\sin[(2a+b)x]}{4(2a+b)} \frac{2 a}{3} \left({x-a}\right)^{3/2} +\frac{2 }{5}\left( {x-a}\right)^{5/2},\text{ or} – \frac{\cos bx}{2b} \text{ where erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt \dfrac{n}{a}\int x^{n-1}e^{ax}\hspace{1pt}\text{d}x \end{equation}, \begin{equation} \int x^2 \sin x\ dx = \left(2-x^2\right) \cos x + 2 x \sin x 19. \end{equation}, \begin{equation}\label{eq:Kloeppel} \frac{1}{b^2-a^2}\left[ 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 Example: The clustrmap is Page 13/24. \int\frac{1}{\sqrt{ax^2+bx+c}}\ dx= \int x\sqrt{x-a}\ dx = \int x e^x \cos x\ dx = \frac{1}{2}e^x (x \cos x \frac{b+2ax}{4a}\sqrt{ax^2+bx+c} "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 â¦ \int \frac{1}{\sqrt{a^2 – x^2}}\ dx = \sin^{-1}\frac{x}{a} 25. 104. 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} \left( – 3b^2 + 2 abx + 8 a(c+ax^2) \right) \int \sec x \ dx = \ln | \sec x + \tan x | = 2 \tanh^{-1} \left (\tan \frac{x}{2} \right) \end{equation}, \begin{equation} + 3(b^3-4abc)\ln \left|b + 2ax + 2\sqrt{a}\sqrt{ax^2+bx+c} \right| \right) \end{equation}, \begin{equation} 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| -\frac{b}{a\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} \text{erf}\left(i\sqrt{ax}\right), \end{equation}, \begin{equation} Types of Integrals. \text{ where } \Gamma(a,x)=\int_x^{\infty} t^{a-1}e^{-t}\hspace{2pt}\text{d}t – \frac{\cos[(2a+b)x]}{4(2a+b)} Not to mention their servers gave up the ghost turned into Zombies on 25 March 2015 (Brains! \int \cosh ax\ dx =\frac{1}{a} \sinh ax \end{equation}, \begin{equation} Table of Integrals BASIC FORMS (1) x n dx ! = uv " vdu ! \end{equation}, \begin{equation} \right. \\ \frac{2}{3} x(x-a)^{3/2} – \frac{4}{15} (x-a)^{5/2}, \text{ or} \int \csc^2 ax\ dx = -\frac{1}{a} \cot ax The Table of Integrals, Series, and Products is the major reference source for integrals in the English language. \int (x+a)^n dx = \frac{(x+a)^{n+1}}{n+1}, n\ne -1 103. \frac{1}{a^2 + b^2} \left[ \end{equation}, \begin{equation} 16. The following is a table of formulas of the commonly used Indefinite Integrals. \int \sqrt{\frac{x}{a+x}}\ dx = \sqrt{x(a+x)} +\frac{1}{2}a^2\tan^{-1}\frac{x}{\sqrt{a^2-x^2}} \int (\ln x)^3\ dx = -6 x+x (\ln x)^3-3 x (\ln x)^2+6 x \ln x -a \cos ax \cosh bx + \end{cases} \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 1. â« u n d u = u n + 1 n + 1 + C, n â â 1 â« u n d u = u n + 1 n + 1 + C, n â â 1. = 1 n + 1 x n + 1 (2) 1 x dx ! \end{equation}, \begin{equation} These restrictions are shown in the third column. (x+a)ndx=(x+a)n a 1+n + x 1+n " #$ % &', n! \int \frac{1}{(x+a)(x+b)}dx = \frac{1}{b-a}\ln\frac{a+x}{b+x}, \text{ } a\ne b \int \sec^2 ax\ dx = \frac{1}{a} \tan ax \frac{2}{15 a^2}(-2b^2+abx + 3 a^2 x^2) 106. \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}\label{eq:Weems} \int x \sqrt{ax + b}\ dx = \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int e^x \cos x\ dx = \frac{1}{2}e^x (\sin x + \cos x) \sqrt{ax+b} \end{equation}, \begin{equation} Table of Basic Integrals Basic Forms There have been visitors to integral-table.com since 2004. \int x^n \cos ax \ dx = Forms â¦ \end{equation}, $$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2}} = \sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+ \infty}_{- \infty}x^{2n} e^{-ax^{2}} = (-1)^{n} \frac{\partial^{n}}{\partial a^{n}}\sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2} + bx} = e^{\frac{b^2}{4a}}\sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \sin^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 – \frac{6(-1)^n}{n^2 \pi^2} \right)$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \cos^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 + \frac{6(-1)^n}{n^2 \pi^2} \right)$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} xÂ \cos \left( \frac{ \pi x}{a} \right)Â \sin \left( \frac{2 \pi x}{a} \right) = \frac{8a^2}{9 \pi ^2}$$, $$\int\limits^{a}_{b} \frac{dx}{\sqrt{\left(a-x \right) \left(x-b \right)}} = \pi \text{ for a > b}$$, $$\int\limits^{a}_{b} \frac{dx}{x\sqrt{\left(a-x \right) \left(x-b \right)}} = \frac{ \pi}{\sqrt{ab}} \text{ for a > b > 0}$$, $$\int\limits^{\frac{\pi}{2}}_{- \frac{\pi}{2}} \frac{dx}{1+ y \sin x} = \frac{\pi}{\sqrt{1 – y^2}} \text{ for -1 < y < 1}$$, $$\int \frac{dx}{\sqrt{a^{2} – x^{2}}} = \text{arcsin} \, \frac{x}{a}$$, $$\int \frac{x dx}{\sqrt{a^{2} + x^{2}}} = \sqrt{a^{2} + x^{2}}$$, $$\int \frac{dx}{\sqrt{a^{2} +x^{2}}} = \text{ln} \, \left(x + \sqrt{a^{2} + x^{2}} \right)$$, $$\int \frac{dx}{a^{2} +x^{2}} = \frac{1}{a} \, \text{arctan} \, \frac{x}{a}$$, $$\int \frac{dx}{ \left( a^{2} + x^{2} \right)^{\frac{3}{2}}} = \frac{1}{a^{2}} \frac{x}{\sqrt{a^{2} +x^{2}}}$$, $$\int\frac{x \, dx}{ \left( a^{2}+x^{2} \right)^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{a^{2} + x^{2}}}$$, $$\int \frac{dx}{\sqrt{ (x – a)^{2} + b^{2}}} = \text{ln} \, \frac{1}{(a – x) + \sqrt{(a-x)^{2} + b^{2}}}$$, $$\int \frac{(x – a) \, dx}{\left[ (x-a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{(x-a)^{2} + b^{2}}}$$, $$\int \frac{dx}{\left[ (x – a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \frac{x – a}{b^{2} \sqrt{(x – a)^{2} +b^{2}}}$$. 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