\int \frac{x^2}{\sqrt{x^2 \pm a^2}}\ dx = \frac{1}{2}x\sqrt{x^2 \pm a^2} (3x2 + 4)d dx {u} = 12 u.u d dx { 2 â 4x2 + 7x5} = 1 2 2 â 4x2 + 7x5 (â8x + 35x4) d dx {c} = 0 , c is a constant ddx {6} = 0 , since â 3.14 is a constant. It is a compilation of the most commonly used integrals. 111. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} Not to mention their servers \int x \cos ax \ dx = \frac{1}{a^2} \cos ax + \frac{x}{a} \sin ax \int \cos^2 ax \sin ax\ dx = -\frac{1}{3a}\cos^3{ax} \int \ln ( x^2 + a^2 )\hspace{.5ex} {dx} = x \ln (x^2 + a^2 ) +2a\tan^{-1} \frac{x}{a} – 2x +\frac{i\sqrt{\pi}}{2a^{3/2}} 4. \frac{\cos[(a+b)x]}{2(a+b)} , a\ne b [Note that you may need to use more than one of the above rules for one integral]. Integral tables >> Basic forms. \end{equation}, \begin{equation} \int x^2 e^{x}\ dx = \left(x^2 – 2x + 2\right) e^{x} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \sqrt{\frac{x}{a+x}}\ dx = \sqrt{x(a+x)} \end{equation}, \begin{equation} The Table of Integrals, Series, and Products is the major reference source for integrals in the English language. 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 \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:Winokur2} 22. \int \frac{1}{\sqrt{a-x}}\ dx = -2\sqrt{a-x} \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:dewitt} \end{equation}, \begin{equation}\label{eq:ritzert} 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. The table presents a selection of integrals found in the Calculus books. \int x^2 \sin ax\ dx =\frac{2-a^2x^2}{a^3}\cos ax +\frac{ 2 x \sin ax}{a^2} 110. >> Table of Basic Integrals Basic Forms There have been visitors to integral-table.com since 2004. + 3(b^3-4abc)\ln \left|b + 2ax + 2\sqrt{a}\sqrt{ax^2+bx+c} \right| \right) Z xndx= xn+1 n+1 +C (n6= 1) 2. \end{equation}, \begin{equation} \int \sin ax \cosh bx \ dx = \end{equation}, \begin{equation} \int x \sqrt{ax + b}\ dx = \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} -\Gamma(n+1, ixa)\right] \int \tan^2 ax\ dx = -x + \frac{1}{a} \tan ax \end{equation}, \begin{equation} \int \tanh ax\hspace{1.5pt} dx =\frac{1}{a} \ln \cosh ax \int \frac{1}{a^2+x^2}dx = \frac{1}{a}\tan^{-1}\frac{x}{a} Table of Trig Integrals. Forms containing trigonometric functions. 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. +\frac{\sin 2bx}{8b}- 28. Free math lessons and math homework help from basic math to algebra, geometry and beyond. \int x e^x\ dx = (x-1) e^x \int \sqrt{a^2 – x^2}\ dx = \frac{1}{2} x \sqrt{a^2-x^2} \end{equation}, \begin{equation} Types of Integrals. \int \sqrt{x(ax+b)}\ dx = \frac{1}{4a^{3/2}}\left[(2ax + b)\sqrt{ax(ax+b)} -\frac{1}{2}(i)^{n+1}\left [ \Gamma(n+1, -ix) \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ a \cosh bx – b \sinh bx ] & a\ne b \\ \end{cases} \int e^x \sin x \ dx = \frac{1}{2}e^x (\sin x – \cos x) \int \tan^3 ax dx = \frac{1}{a} \ln \cos ax + \frac{1}{2a}\sec^2 ax \int x^2 \cos x \ dx = 2 x \cos x + \left ( x^2 – 2 \right ) \sin x \int \cos ax \cosh bx\ dx = 109. \displaystyle{ \frac{ e^{(a+2b)x}}{(a+2b)} A: TABLE OF BASIC DERIVATIVES Let u = u(x) be a differentiable function of the independent variable x, that is u(x) exists. The clustrmap is Page 13/24. \mp \frac{1}{2}a^2 \ln \left| x + \sqrt{x^2\pm a^2} \right | Sometimes restrictions need to be placed on the values of some of the variables. – \sin x + x \sin x) \int x^2 \ln x \ dx = \frac{1}{3} x^3 \ln x-\frac{x^3}{9} -\frac{\sin[(2a-b)x]}{4(2a-b)} – \frac{\cos[(2a+b)x]}{4(2a+b)} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \cos^p ax dx = -\frac{1}{a(1+p)}{\cos^{1+p} ax} \times It is essential for mathematicians, scientists, and engineers, who rely on it when identifying and subsequently solving extremely complex problems. \int \sin^3 ax \ dx = -\frac{3 \cos ax}{4a} + \frac{\cos 3ax} {12a} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \right. \int x \ln (ax + b)\ dx = \frac{bx}{2a}-\frac{1}{4}x^2 \int \frac{x^3}{a^2+x^2}dx = \frac{1}{2}x^2-\frac{1}{2}a^2\ln|a^2+x^2| \int x \sin^2 x \ dx = \frac{x^2}{4}-\frac{1}{8}\cos 2x – \frac{1}{4} x \sin 2x /Filter /FlateDecode Forms containing logarithms and exponentials. \end{equation}, \begin{equation} \frac{1}{a^2 + b^2} \left[ \end{equation}, \begin{equation}\label{eq:Rigo} \int x^n \sin x \ dx = -\frac{1}{2}(i)^n\left[ \Gamma(n+1, -ix) \end{equation}, \begin{equation} 1. \end{equation}, \begin{equation} b \cosh bx \sinh ax \right] \end{equation}, \begin{equation}\label{eq:Duley} \int x\sqrt{x-a}\ dx = \end{equation}, \begin{equation}\label{eq:Gilmore} \dfrac{n}{a}\int x^{n-1}e^{ax}\hspace{1pt}\text{d}x \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| 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. \int \frac{1}{\sqrt{a^2 – x^2}}\ dx = \sin^{-1}\frac{x}{a} \int \sqrt{x} e^{ax}\ dx = \frac{1}{a}\sqrt{x}e^{ax} \sqrt{ax+b} Table of Basic Integrals Basic Forms 1 Z (1) xn dx = xn+1 , n 6= â1 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= â1 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 = tanâ1 x 1 + x2 1 1 Z â1 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} \right] \int x \cos x \ dx = \cos x + x \sin x = uv " vdu ! \sqrt{x^3(ax+b)} + – \frac{\sin[(2a+b)x]}{4(2a+b)} -\frac{1}{a}{\cos ax} \hspace{2mm}{_2F_1}\left[ \end{equation}, \begin{equation}\label{eq:swift3} \int x^3 e^{x}\ dx = \left(x^3-3x^2 + 6x – 6\right) e^{x} \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ -b \cosh bx + a \sinh bx ] & a\ne b \\ \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \ln ax\ dx = x \ln ax – x \end{equation}, \begin{equation} \int \sin^2 ax\ dx = \frac{x}{2} – \frac{\sin 2ax} {4a} \end{equation}, \begin{equation} 7 0 obj << \int x \sin ax\ dx = -\frac{x \cos ax}{a} + \frac{\sin ax}{a^2} \displaystyle{\frac{e^{ax}-2\tan^{-1}[e^{ax}]}{a} } & a = b \int \frac{x}{(x+a)^2}dx = \frac{a}{a+x}+\ln |a+x| \int \sec^2 x \tan x\ dx = \frac{1}{2} \sec^2 x Basic Integrals. \right] 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. \end{equation}, \begin{equation} 31. ���_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*��>\�+�. Basic Integrals. \int \sin^2 ax \cos^2 bx dx = \frac{x}{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 {_2}F_1\left( \frac{n+1}{2}, \int \sin ax \sinh bx \ dx = Notify me of follow-up comments by email. \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:Kloeppel} \int e^{ax} \cosh bx \ dx = \right] \end{equation}, \begin{equation} \int &x \sqrt{a x^2 + bx + c}\ dx = \frac{1}{48a^{5/2}}\left ( \frac{\tan^{n+1} ax }{a(1+n)} \times \int \cos^2 ax\ dx = \frac{x}{2}+\frac{ \sin 2ax}{4a} \int x^2 e^{ax}\ dx = \left(\frac{x^2}{a}-\frac{2x}{a^2}+\frac{2}{a^3}\right) e^{ax} 107. 99. \end{equation}, \begin{equation} \int \frac{1}{1+x^2}dx = \tan^{-1}x 102. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} You can verify any of the formulas by differentiating the function on the right side and obtaining the integrand. \begin{cases} \int \sec x \tan x\ dx = \sec x \int x \ln \left ( a^2 – b^2 x^2 \right )\ dx = -\frac{1}{2}x^2+ \int x^n \cos ax \ dx = \int e^{ax}\ dx = \frac{1}{a}e^{ax} \end{equation}, \begin{equation}\label{eq:qarles2} \frac{b}{2a^{3/2}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right | \\ & \left. \end{equation}, \begin{equation} Integration â is one of the main mathematical operations. \end{equation}, \begin{equation} 101. \end{equation}, \begin{equation} Administrator of Mini Physics. \right . +\frac{1}{2}a^2\tan^{-1}\frac{x}{\sqrt{a^2-x^2}} \int\frac{dx}{(a^2+x^2)^{3/2}}=\frac{x}{a^2\sqrt{a^2+x^2}} \int \sec x \csc x \ dx = \ln | \tan x | \end{equation}, \begin{equation} \int x^n \cos x dx = \int e^{ax} \sinh bx \ dx = \int \frac{x}{a^2+x^2}dx = \frac{1}{2}\ln|a^2+x^2| \end{equation}, \begin{equation} \int\frac{1}{\sqrt{ax^2+bx+c}}\ dx= } View Notes - Table_of_Integrals from MAP 3305 at Florida Atlantic University. (x+a)ndx=(x+a)n a 1+n + x 1+n " #$ % &', n! \int \frac{x^2}{a^2+x^2}dx = x-a\tan^{-1}\frac{x}{a} \end{split} \frac{1}{a^2 + b^2} \left[ \end{equation}, \begin{equation} + (-1)^n \Gamma(n+1, ix)\right] \end{equation}, \begin{equation} \int (\ln x)^2\ dx = 2x – 2x \ln x + x (\ln x)^2 \int x^n e^{ax}\ dx = \frac{(-1)^n}{a^{n+1}}\Gamma[1+n,-ax], \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:swift1} \end{equation}, \begin{equation} – \frac{\cos bx}{2b} \\ \frac{2}{3} x(x-a)^{3/2} – \frac{4}{15} (x-a)^{5/2}, \text{ or} \end{equation}, \begin{equation} \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 \csc^3 x\ dx = -\frac{1}{2}\cot x \csc x + \frac{1}{2} \ln | \csc x – \cot x | \end{equation}, \begin{equation} \end{equation}, $$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2}} = \sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+ \infty}_{- \infty}x^{2n} e^{-ax^{2}} = (-1)^{n} \frac{\partial^{n}}{\partial a^{n}}\sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2} + bx} = e^{\frac{b^2}{4a}}\sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \sin^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 – \frac{6(-1)^n}{n^2 \pi^2} \right)$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \cos^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 + \frac{6(-1)^n}{n^2 \pi^2} \right)$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x \cos \left( \frac{ \pi x}{a} \right) \sin \left( \frac{2 \pi x}{a} \right) = \frac{8a^2}{9 \pi ^2} $$, $$\int\limits^{a}_{b} \frac{dx}{\sqrt{\left(a-x \right) \left(x-b \right)}} = \pi \text{ for a > b}$$, $$\int\limits^{a}_{b} \frac{dx}{x\sqrt{\left(a-x \right) \left(x-b \right)}} = \frac{ \pi}{\sqrt{ab}} \text{ for a > b > 0}$$, $$\int\limits^{\frac{\pi}{2}}_{- \frac{\pi}{2}} \frac{dx}{1+ y \sin x} = \frac{\pi}{\sqrt{1 – y^2}} \text{ for -1 < y < 1}$$, $$\int \frac{dx}{\sqrt{a^{2} – x^{2}}} = \text{arcsin} \, \frac{x}{a}$$, $$\int \frac{x dx}{\sqrt{a^{2} + x^{2}}} = \sqrt{a^{2} + x^{2}}$$, $$\int \frac{dx}{\sqrt{a^{2} +x^{2}}} = \text{ln} \, \left(x + \sqrt{a^{2} + x^{2}} \right)$$, $$\int \frac{dx}{a^{2} +x^{2}} = \frac{1}{a} \, \text{arctan} \, \frac{x}{a}$$, $$\int \frac{dx}{ \left( a^{2} + x^{2} \right)^{\frac{3}{2}}} = \frac{1}{a^{2}} \frac{x}{\sqrt{a^{2} +x^{2}}}$$, $$\int\frac{x \, dx}{ \left( a^{2}+x^{2} \right)^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{a^{2} + x^{2}}}$$, $$\int \frac{dx}{\sqrt{ (x – a)^{2} + b^{2}}} = \text{ln} \, \frac{1}{(a – x) + \sqrt{(a-x)^{2} + b^{2}}}$$, $$\int \frac{(x – a) \, dx}{\left[ (x-a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{(x-a)^{2} + b^{2}}}$$, $$\int \frac{dx}{\left[ (x – a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \frac{x – a}{b^{2} \sqrt{(x – a)^{2} +b^{2}}}$$. Table of Integrals Engineers usually refer to a table of integrals when performing calculations involving integration. 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 ⦠\int \sec^3 x \ {dx} = \frac{1}{2} \sec x \tan x + \frac{1}{2}\ln | \sec x + \tan x | \int \csc^2 ax\ dx = -\frac{1}{a} \cot ax Forms ⦠\int \frac{1}{x}dx = \ln |x| Home University Mathematics Integration Table, \begin{equation} \int e^x \cos x\ dx = \frac{1}{2}e^x (\sin x + \cos x) \int x \cos^2 x \ dx = \frac{x^2}{4}+\frac{1}{8}\cos 2x + \frac{1}{4} x \sin 2x \end{equation}, \begin{equation} \pm\frac{1}{2}a^2 \ln \left | x + \sqrt{x^2\pm a^2} \right | \int \sin^2 x \cos x\ dx = \frac{1}{3} \sin^3 x Indefinite integrals. \int \frac{1}{ax+b}dx = \frac{1}{a} \ln |ax + b| \end{equation}, \begin{equation} & a\ne b \\ Free Table of Integrals to print on a single sheet side and side. \end{equation}, \begin{equation} \end{equation}, \begin{equation} x��ZIs�F��W�V�v�KR9$�qj\SS5c�e�h \int \sinh ax \cosh ax dx= = 1 n + 1 x n + 1 (2) 1 x dx ! 1, \frac{n+3}{2}, -\tan^2 ax \right) \frac{x}{3}\right] \int \cos^2 ax \sin bx\ dx = \frac{\cos[(2a-b)x]}{4(2a-b)} u ddx {(x3 + 4x + 1)3/4} = 34 (x3 + 4x + 1)â1/4. 19. -\frac{b}{a\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} \end{equation}, \begin{equation} -b^2 \ln \left| a\sqrt{x} + \sqrt{a(ax+b)} \right| \right ] = ln x (3) udv ! 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 \int \sin ax \ dx = -\frac{1}{a} \cos ax \end{equation}, \begin{equation}\label{eq:Larry-Morris}\begin{split} \end{equation}, \begin{equation}\label{eq:ebke} \end{equation}, \begin{equation} \end{equation}, \begin{equation} /Length 2403 \begin{array}{l} \end{equation}, \begin{equation} 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. 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