\end{equation}, \begin{equation} \int \cos^2 ax\ dx = \frac{x}{2}+\frac{ \sin 2ax}{4a} It is a compilation of the most commonly used integrals. Note: Most of the following integral entries are written for indefinite integrals, but they also apply to definite integrals. \int x^n \sin x \ dx = -\frac{1}{2}(i)^n\left[ \Gamma(n+1, -ix) 110. \int x \tan^2 x \ dx = -\frac{x^2}{2} + \ln \cos x + x \tan x \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \sec^2 x \tan x\ dx = \frac{1}{2} \sec^2 x \end{equation}, \begin{equation} \end{equation}, \begin{equation} + (-1)^n \Gamma(n+1, ix)\right] \end{equation}, \begin{equation}\label{eq:xul} \int \frac{x}{ax^2+bx+c}dx = \frac{1}{2a}\ln|ax^2+bx+c| 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 \int \sin^2 ax \cos^2 bx dx = \frac{x}{4} The following is a table of formulas of the commonly used Indefinite Integrals. \\ & \left. 112. \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:dewitt} a \cos ax \sinh bx \int \frac{\ln ax}{x}\ dx = \frac{1}{2}\left ( \ln ax \right)^2 -2x + \left( \frac{b}{2a}+x \right )\ln \left (ax^2+bx+c \right) \int e^{ax}\ dx = \frac{1}{a}e^{ax} Use the table of integral formulas and the rules above to evaluate the following integrals. \end{equation}, \begin{equation} All the immediate integrals. 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| Table of Standard Integrals 1. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int x \ln (ax + b)\ dx = \frac{bx}{2a}-\frac{1}{4}x^2 \displaystyle{ \int \frac{1}{\sqrt{x\pm a}}\ dx = 2\sqrt{x\pm a} \end{equation}, \begin{equation} \int e^{bx} \sin ax\ dx = \frac{1}{a^2+b^2}e^{bx} (b\sin ax – a\cos ax) \end{equation}, \begin{equation} \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ a \cosh bx – b \sinh bx ] & a\ne b \\ \int x e^{-ax^2}\ {dx} = -\dfrac{1}{2a}e^{-ax^2} \end{cases} \end{equation}, \begin{equation} Forms containing logarithms and exponentials. Free Integration Worksheet. (x+a)ndx=(x+a)n a 1+n + x 1+n " #$ % &', n! +\frac{\sin 2bx}{8b}- \int (\ln x)^2\ dx = 2x – 2x \ln x + x (\ln x)^2 \int x \cos ax \ dx = \frac{1}{a^2} \cos ax + \frac{x}{a} \sin ax \displaystyle{\frac{e^{2ax}}{4a} + \frac{x}{2}} & a = b \end{equation}, \begin{equation} \int \sqrt{x-a}\ dx = \frac{2}{3}(x-a)^{3/2} \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 \end{equation}, \begin{equation} \int \frac{1}{x}dx = \ln |x| \int e^{bx} \cos ax\ dx = \frac{1}{a^2 + b^2} e^{bx} ( a \sin ax + b \cos ax ) The clustrmap is Page 13/24. 10. \end{equation}, \begin{equation} 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 -\Gamma(n+1, ixa)\right] \int x^2 \sin x\ dx = \left(2-x^2\right) \cos x + 2 x \sin x 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 ⦠Basic Differentiation Rules Basic Integration Formulas DERIVATIVES AND INTEGRALS © Houghton Mifflin Company, Inc. 1. \int x \cos x \ dx = \cos x + x \sin x \int \frac{1}{(x+a)(x+b)}dx = \frac{1}{b-a}\ln\frac{a+x}{b+x}, \text{ } a\ne b \int u dv = uv – \int v du \frac{1}{4a}\left[ 106. \end{equation}, \begin{equation} \int \cos ax \sinh bx\ dx = 105. + \frac{\sin bx}{2b} \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:Weems} \int \frac{x}{(x+a)^2}dx = \frac{a}{a+x}+\ln |a+x| Sometimes restrictions need to be placed on the values of some of the variables. Table of integrals - the basic formulas of indefinite integrals. \end{equation}, \begin{equation}\label{eq:Duley} {_2F_1}\left[ 1+\frac{a}{2b},1,2+\frac{a}{2b}, -e^{2bx}\right] }& \\ – \frac{\cos[(2a+b)x]}{4(2a+b)} b \sin ax \sinh bx 7 0 obj << \end{equation}, \begin{equation} \end{equation}, \begin{equation} \displaystyle{ \frac{ e^{(a+2b)x}}{(a+2b)} [latex]\int {u}^{n}du=\frac{{u}^{n+1}}{n+1}+C,n\ne \text{â}1[/latex] 2. \end{equation}, \begin{equation} \int \cos^2 ax \sin bx\ dx = \frac{\cos[(2a-b)x]}{4(2a-b)} /Length 2403 b \cosh bx \sinh ax -\frac{b}{a\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} \int \frac{1}{ax+b}dx = \frac{1}{a} \ln |ax + b| Example: \begin{cases} \begin{cases} \int x \sec^2 x \ dx = \ln \cos x + x \tan x \end{equation}, \begin{equation} Tinycards by Duolingo is a fun flashcard app that helps you memorize anything for free, forever. \end{equation}, \begin{equation}\label{eq:Winokur1} ��H�$e���� �XH*N�"���뷿�u7M>$4��������kffgJ&��N9�N'�jB�Mn�ۅ����C�ȄQ��}����n�*��Y�����a����� � \\ \frac{2}{15}(2a+3x)(x-a)^{3/2} \right. If you spot any errors or want to suggest improvements, please contact us. \right] \int \sin^3 ax \ dx = -\frac{3 \cos ax}{4a} + \frac{\cos 3ax} {12a} \int \sin ax \cosh bx \ dx = \int \tanh ax\hspace{1.5pt} dx =\frac{1}{a} \ln \cosh ax \end{equation}, \begin{equation} u ddx {(x3 + 4x + 1)3/4} = 34 (x3 + 4x + 1)â1/4. (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. Table of Integrals BASIC FORMS (1) x n dx ! = ln x (3) udv ! \int \sqrt{x(ax+b)}\ dx = \frac{1}{4a^{3/2}}\left[(2ax + b)\sqrt{ax(ax+b)} \frac{1}{2}, \frac{1-n}{2}, \frac{3}{2}, \cos^2 ax These restrictions are shown in the third column. \frac{1+p}{2}, \frac{1}{2}, \frac{3+p}{2}, \cos^2 ax \int \sqrt{\frac{x}{a-x}}\ dx = -\sqrt{x(a-x)} \int x \cos^2 x \ dx = \frac{x^2}{4}+\frac{1}{8}\cos 2x + \frac{1}{4} x \sin 2x 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. \int \frac{1}{ax^2+bx+c}dx = \frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}} 100. \int x^2 \cos x \ dx = 2 x \cos x + \left ( x^2 – 2 \right ) \sin x Integration is the basic operation in integral calculus. \pm\frac{1}{2}a^2 \ln \left | x + \sqrt{x^2\pm a^2} \right | {_2F_1}\left[ \end{equation}, \begin{equation} This leaï¬et provides such a table. \frac{1}{a}\sqrt{ax^2+bx + c} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int e^{ax} \tanh bx\ dx = \end{equation}, \begin{equation} + \int \sin ax \sinh bx \ dx = \end{equation}, \begin{equation} \end{equation}, \begin{equation} \frac{1}{a^2 + b^2} \left[ \\ \frac{2}{3} x(x-a)^{3/2} – \frac{4}{15} (x-a)^{5/2}, \text{ or} Notify me of follow-up comments by email. Forms containing inverse trigonometric functions. Z xndx= xn+1 n+1 +C (n6= 1) 2. \int \sin^2 ax\ dx = \frac{x}{2} – \frac{\sin 2ax} {4a} 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) \int \sec^n x \tan x \ dx = \frac{1}{n} \sec^n x , n\ne 0 \int \sec x \tan x\ dx = \sec x View Notes - Table_of_Integrals from MAP 3305 at Florida Atlantic University. 28. \int &x \sqrt{a x^2 + bx + c}\ dx = \frac{1}{48a^{5/2}}\left ( \int \tan^3 ax dx = \frac{1}{a} \ln \cos ax + \frac{1}{2a}\sec^2 ax \int x \sin^2 x \ dx = \frac{x^2}{4}-\frac{1}{8}\cos 2x – \frac{1}{4} x \sin 2x \right] \int x^2 e^{ax}\ dx = \left(\frac{x^2}{a}-\frac{2x}{a^2}+\frac{2}{a^3}\right) e^{ax} – \frac{\sin[(2a+b)x]}{4(2a+b)} \int \csc^3 x\ dx = -\frac{1}{2}\cot x \csc x + \frac{1}{2} \ln | \csc x – \cot x | \int x(x+a)^n dx = \frac{(x+a)^{n+1} ( (n+1)x-a)}{(n+1)(n+2)} \end{equation}, \begin{equation} On this page, the tables contain examples of the most common integrals. Free math lessons and math homework help from basic math to algebra, geometry and beyond. \int \cos^p ax dx = -\frac{1}{a(1+p)}{\cos^{1+p} ax} \times \end{cases} \end{equation}, \begin{equation} \int \frac{\ln x}{x^2}\ dx = -\frac{1}{x}-\frac{\ln x}{x} \int e^{ax} \cosh bx \ dx = Indefinite integrals. \displaystyle{\frac{e^{2ax}}{4a} – \frac{x}{2}} & a = b 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. \int \sec^3 x \ {dx} = \frac{1}{2} \sec x \tan x + \frac{1}{2}\ln | \sec x + \tan x | [Note that you may need to use more than one of the above rules for one integral]. \int x \ln x \ dx = \frac{1}{2} x^2 \ln x-\frac{x^2}{4} \end{equation}, \begin{equation} \int \sin^2 x \cos x\ dx = \frac{1}{3} \sin^3 x \end{equation}, \begin{equation} 101. \text{ where erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt = 1 n + 1 x n + 1 (2) 1 x dx ! \int \sqrt{ax+b}\ dx = \left(\frac{2b}{3a}+\frac{2x}{3}\right)\sqrt{ax+b} \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 +\frac{1}{2}\left(x^2-\frac{b^2}{a^2}\right)\ln (ax+b) 22. \end{equation}, \begin{equation} \frac{1}{a^2 + b^2} \left[ Integrals Involving a + bu, a â 0. \int x^n \cos ax \ dx = Basic Integrals. \int \csc x\ dx = \ln \left | \tan \frac{x}{2} \right| = \ln | \csc x – \cot x| + C 107. \end{equation}, \begin{equation} \frac{4ac-b^2}{8a^{3/2}}\ln \left| 2ax + b + 2\sqrt{a(ax^2+bx^+c)}\right | 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. Integral Table. \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 [latex]\int \frac{du}{u}=\text{ln}|u|+C[/latex] 3. \end{equation}, \begin{equation}\label{eq:veky} -\frac{\sin[(2a-b)x]}{4(2a-b)} \int (x+a)^n dx = \frac{(x+a)^{n+1}}{n+1}, n\ne -1 + 3(b^3-4abc)\ln \left|b + 2ax + 2\sqrt{a}\sqrt{ax^2+bx+c} \right| \right) \frac{b^3}{8a^{5/2}}\ln \left | a\sqrt{x} + \sqrt{a(ax+b)} \right | \end{equation}, \begin{equation} \int \sin^n ax \ dx = \end{equation}, \begin{equation}\label{eq:Russ} \int \sinh ax \cosh ax dx= \int x^n e^{ax}\ dx = \dfrac{x^n e^{ax}}{a} – \frac{1}{a^2 + b^2} \left[ \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ -b \cosh bx + a \sinh bx ] & a\ne b \\ \end{equation}, \begin{equation} \int \ln ( x^2 – a^2 )\hspace{.5ex} {dx} = x \ln (x^2 – a^2 ) +a\ln \frac{x+a}{x-a} – 2x \int e^{-ax^2}\ dx = \frac{\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(x\sqrt{a}\right) \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. \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} \end{equation}, \begin{equation} \frac{x}{3}\right] \int \frac{x^2}{\sqrt{x^2 \pm a^2}}\ dx = \frac{1}{2}x\sqrt{x^2 \pm a^2} \int x \sin x\ dx = -x \cos x + \sin x 1, \frac{n+3}{2}, -\tan^2 ax \right) Free Table of Integrals to print on a single sheet side and side. \int x^n e^{ax}\ dx = \frac{(-1)^n}{a^{n+1}}\Gamma[1+n,-ax], Basic forms. – (-1)^n\Gamma(n+1, -ix)\right] \int \cos ax\ dx= \frac{1}{a} \sin ax Table of Integrals. Basic Integrals. 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 \int \sin^2 ax \cos bx\ dx = \end{equation}, \begin{equation} \int x^n \cos x dx = \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}} 98. \end{equation}, \begin{equation}\label{eq:ebke} \end{equation}, \begin{equation} -\frac{\sin 2ax}{8a}- \frac{1}{a^2 + b^2} \left[ -2ax + \sinh 2ax \right] \end{equation}, \begin{equation} \end{equation}, \begin{equation} \left( – 3b^2 + 2 abx + 8 a(c+ax^2) \right) \hspace{1cm}-\frac{1}{a}e^{ax}{_2F_1}\left[ 1, \frac{a}{2b},1+\frac{a}{2b}, -e^{2bx}\right] Forms ⦠– \end{equation}, \begin{equation} It is essential for mathematicians, scientists, and engineers, who rely on it when identifying and subsequently solving extremely complex problems. Not to mention their servers gave up the ghost turned into Zombies on 25 March 2015 (Brains! 4)>$�ÿ�K��1��~)���$��z!~Z��dBPb�H2͈к$��*��'�z�E���D�S#J���t�u�aլM��$.1�����8Q���q3Ds�d-���YOeU)(h��$ �Dp�XBm� \end{equation}, \begin{equation} \int x \sin ax\ dx = -\frac{x \cos ax}{a} + \frac{\sin ax}{a^2} \int \sec x \csc x \ dx = \ln | \tan x | \int \frac{1}{\sqrt{a-x}}\ dx = -2\sqrt{a-x} 19. \frac{\sin[2(a+b)x]}{16(a+b)} \end{equation}, \begin{equation} Table of Basic Integrals Basic Forms There have been visitors to integral-table.com since 2004. \end{equation}, \begin{equation} \end{equation}, \begin{equation} 113. \int \sec^2 ax\ dx = \frac{1}{a} \tan ax \int \tan^2 ax\ dx = -x + \frac{1}{a} \tan ax \int x e^x \sin x\ dx = \frac{1}{2}e^x (\cos x – x \cos x + x \sin x) \end{equation}, \begin{equation} Table of Integrals Engineers usually refer to a table of integrals when performing calculations involving integration. a \sin ax \cosh bx + b \cos ax \sinh bx \end{equation}, \begin{equation} \int \frac{x}{\sqrt{a^2-x^2}}\ dx = -\sqrt{a^2-x^2} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int x\sqrt{x-a}\ dx = \int \frac{x}{\sqrt{x^2\pm a^2}}\ dx = \sqrt{x^2 \pm a^2} \right] \int \csc^nx \cot x\ dx = -\frac{1}{n}\csc^n x, n\ne 0 109. Table of Indefinite Integral Formulas . 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} You can verify any of the formulas by differentiating the function on the right side and obtaining the integrand. \end{equation}, \begin{equation} \int\frac{1}{\sqrt{ax^2+bx+c}}\ dx= \int\frac{dx}{(a^2+x^2)^{3/2}}=\frac{x}{a^2\sqrt{a^2+x^2}} ���_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*��>\�+�. \text{ where } \Gamma(a,x)=\int_x^{\infty} t^{a-1}e^{-t}\hspace{2pt}\text{d}t -a \cos ax \cosh bx + \frac{1}{2}(ia)^{1-n}\left [ (-1)^n \Gamma(n+1, -iax) \int (ax+b)^{3/2}\ dx =\frac{2}{5a}(ax+b)^{5/2} \end{equation}, \begin{equation} \int \sqrt{a x^2 + b x + c}\ dx = \end{equation}, \begin{equation}\label{eq:qarles1} %PDF-1.5 \frac{1}{b^2-a^2}\left[ a \sin ax \sinh bx Integration â is one of the main mathematical operations. \end{equation}, \begin{equation} \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} \int \cosh ax\ dx =\frac{1}{a} \sinh ax \end{equation}, \begin{equation} \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} \end{equation}, \begin{equation} 104. 99. \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:Winokur2} \right] \int \sin ax \ dx = -\frac{1}{a} \cos ax -b^2 \ln \left| a\sqrt{x} + \sqrt{a(ax+b)} \right| \right ] \frac{2}{15 a^2}(-2b^2+abx + 3 a^2 x^2) 1. 2. \int \frac{1}{\sqrt{a^2 – x^2}}\ dx = \sin^{-1}\frac{x}{a} \end{equation}, \begin{equation}\label{eq:Gilmore} Represent constants examples: d dx { un } = 34 ( x3 + 4x + 1 3/4... 1 x n dx for integrals in the Calculus books the above rules for one integral table of basic integrals... 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