 # table of basic integrals

\text{erf}\left(i\sqrt{ax}\right), \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:xul} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} ÑÐ°Ð±Ð»Ð¸ÑÐ° Ð¸Ð½ÑÐµÐ³ÑÐ°Ð»Ð¾Ð². \int x e^x \cos x\ dx = \frac{1}{2}e^x (x \cos x \int \sin^2 ax \cos^2 bx dx = \frac{x}{4} \frac{1}{a^2 + b^2} \left[ -a \cos ax \cosh bx + \end{equation}, \begin{equation} \frac{b}{12a}- 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 \end{cases} 99. \displaystyle{\frac{e^{2ax}}{4a} + \frac{x}{2}} & a = b \int \frac{x^3}{a^2+x^2}dx = \frac{1}{2}x^2-\frac{1}{2}a^2\ln|a^2+x^2| \right] \int x^2 \sin ax\ dx =\frac{2-a^2x^2}{a^3}\cos ax +\frac{ 2 x \sin ax}{a^2} \int \sin^2 ax\ dx = \frac{x}{2} – \frac{\sin 2ax} {4a} \right. Apr 30, 2018 - Complete table of integrals in a single sheet. Forms containing trigonometric functions. 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{cases} 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\frac{1}{\sqrt{ax^2+bx+c}}\ dx= \int x^n \sin x \ dx = -\frac{1}{2}(i)^n\left[ \Gamma(n+1, -ix) \end{equation}, \begin{equation}\label{eq:Gilmore} \int x^n \cos x dx = \frac{\sin[2(a+b)x]}{16(a+b)} \begin{cases} \end{equation}, \begin{equation} & a\ne b \\ 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 \sec x \csc x \ dx = \ln | \tan x | \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:veky} \end{equation}, \begin{equation} \int e^{ax} \cosh bx \ dx = \int \tan ax\ dx = -\frac{1}{a} \ln \cos ax – \sin x + x \sin x) There have been visitors to integral-table.com since 2004. \end{equation}, \begin{equation} \int \frac{\ln ax}{x}\ dx = \frac{1}{2}\left ( \ln ax \right)^2 \int \sec x \tan x\ dx = \sec x \int \ln ( x^2 – a^2 )\hspace{.5ex} {dx} = x \ln (x^2 – a^2 ) +a\ln \frac{x+a}{x-a} – 2x \frac{1}{2}\left( x^2 – \frac{a^2}{b^2} \right ) \ln \left (a^2 -b^2 x^2 \right) \frac{1}{2}(ia)^{1-n}\left [ (-1)^n \Gamma(n+1, -iax) \end{equation}, \begin{equation} \int x \cos^2 x \ dx = \frac{x^2}{4}+\frac{1}{8}\cos 2x + \frac{1}{4} x \sin 2x \int x(x+a)^n dx = \frac{(x+a)^{n+1} ( (n+1)x-a)}{(n+1)(n+2)} \int e^{bx} \cos ax\ dx = \frac{1}{a^2 + b^2} e^{bx} ( a \sin ax + b \cos ax ) \int x \sqrt{x^2 \pm a^2}\ dx= \frac{1}{3}\left ( x^2 \pm a^2 \right)^{3/2} Table of Integrals Engineers usually refer to a table of integrals when performing calculations involving integration. \int \frac{x}{\sqrt{ax^2+bx+c}}\ dx= \end{equation}, \begin{equation} 4. \end{equation}, \begin{equation}\label{eq:Winokur1} Free Integration Worksheet. Integrals Involving a + bu, a â  0. \end{equation}, \begin{equation} \\ \frac{2}{3} x(x-a)^{3/2} – \frac{4}{15} (x-a)^{5/2}, \text{ or} \int \frac{1}{1+x^2}dx = \tan^{-1}x \int \tan^3 ax dx = \frac{1}{a} \ln \cos ax + \frac{1}{2a}\sec^2 ax \int \sinh ax\ dx = \frac{1}{a} \cosh ax \end{equation}, \begin{equation} \int \cos^3 ax dx = \frac{3 \sin ax}{4a}+\frac{ \sin 3ax}{12a} 22. \end{equation}, \begin{equation} The clustrmap is Page 13/24. Table of Integralsâ. \int e^{ax^2}\ dx = -\frac{i\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(ix\sqrt{a}\right) \displaystyle{\frac{e^{2ax}}{4a} – \frac{x}{2}} & a = b -2x + \left( \frac{b}{2a}+x \right )\ln \left (ax^2+bx+c \right) Forms â¦ \int (x+a)^n dx = \frac{(x+a)^{n+1}}{n+1}, n\ne -1 112. Not to mention their servers A: TABLE OF BASIC DERIVATIVES Let u = u(x) be a differentiable function of the independent variable x, that is u(x) exists. 7 0 obj << – \frac{\sin[(2a+b)x]}{4(2a+b)} {_2F_1}\left[ 1+\frac{a}{2b},1,2+\frac{a}{2b}, -e^{2bx}\right] }& \\ \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, $$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2}} = \sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+ \infty}_{- \infty}x^{2n} e^{-ax^{2}} = (-1)^{n} \frac{\partial^{n}}{\partial a^{n}}\sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2} + bx} = e^{\frac{b^2}{4a}}\sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \sin^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 – \frac{6(-1)^n}{n^2 \pi^2} \right)$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \cos^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 + \frac{6(-1)^n}{n^2 \pi^2} \right)$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} xÂ \cos \left( \frac{ \pi x}{a} \right)Â \sin \left( \frac{2 \pi x}{a} \right) = \frac{8a^2}{9 \pi ^2}$$, $$\int\limits^{a}_{b} \frac{dx}{\sqrt{\left(a-x \right) \left(x-b \right)}} = \pi \text{ for a > b}$$, $$\int\limits^{a}_{b} \frac{dx}{x\sqrt{\left(a-x \right) \left(x-b \right)}} = \frac{ \pi}{\sqrt{ab}} \text{ for a > b > 0}$$, $$\int\limits^{\frac{\pi}{2}}_{- \frac{\pi}{2}} \frac{dx}{1+ y \sin x} = \frac{\pi}{\sqrt{1 – y^2}} \text{ for -1 < y < 1}$$, $$\int \frac{dx}{\sqrt{a^{2} – x^{2}}} = \text{arcsin} \, \frac{x}{a}$$, $$\int \frac{x dx}{\sqrt{a^{2} + x^{2}}} = \sqrt{a^{2} + x^{2}}$$, $$\int \frac{dx}{\sqrt{a^{2} +x^{2}}} = \text{ln} \, \left(x + \sqrt{a^{2} + x^{2}} \right)$$, $$\int \frac{dx}{a^{2} +x^{2}} = \frac{1}{a} \, \text{arctan} \, \frac{x}{a}$$, $$\int \frac{dx}{ \left( a^{2} + x^{2} \right)^{\frac{3}{2}}} = \frac{1}{a^{2}} \frac{x}{\sqrt{a^{2} +x^{2}}}$$, $$\int\frac{x \, dx}{ \left( a^{2}+x^{2} \right)^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{a^{2} + x^{2}}}$$, $$\int \frac{dx}{\sqrt{ (x – a)^{2} + b^{2}}} = \text{ln} \, \frac{1}{(a – x) + \sqrt{(a-x)^{2} + b^{2}}}$$, $$\int \frac{(x – a) \, dx}{\left[ (x-a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{(x-a)^{2} + b^{2}}}$$, $$\int \frac{dx}{\left[ (x – a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \frac{x – a}{b^{2} \sqrt{(x – a)^{2} +b^{2}}}$$. >> \int \cos^2 ax\ dx = \frac{x}{2}+\frac{ \sin 2ax}{4a} ���_eE�j��M���X{�x��4�×oJ����@��p8S9<>$oo�U���{�LrR뾉�눖����E�9OYԚ�X����E��\��� �k�o�r�f�Y��#�j�:�#�x��sƉ�&��R�w��Aj��Dq�d���1t�P����B�wC�D�(ɓ�f�H�"�Ț���HĔ� ���r�0�ZN����.�l2����76}�;L���H�� �ᬦ�cRk��ё(c��+���C�Q�ٙ��tK�eR���9&ׄ�^�X�0l���9��HjNC��Dxԗ)�%tzw��8�u9dKB*��>\�+�. 107. \int \sqrt{a^2 – x^2}\ dx = \frac{1}{2} x \sqrt{a^2-x^2} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \\ & \left. \end{equation}, \begin{equation} -\frac{\sin[(2a-b)x]}{4(2a-b)} ��H�$e���׍� �XH*N�"���뷿�u7M>$4��������kffgJ&��N9�N'�jB�Mn�ۅ����C�ȄQ��}����n�*��Y�����a����� � 31. \end{equation}, \begin{equation} 98. 108. \frac{1}{a}\sqrt{ax^2+bx + c} \end{equation}, \begin{equation} \end{equation}, \begin{equation} This leaï¬et provides such a table. \end{equation}, \begin{equation} \int x^2 \sin x\ dx = \left(2-x^2\right) \cos x + 2 x \sin x \int \frac{1}{a^2+x^2}dx = \frac{1}{a}\tan^{-1}\frac{x}{a} Administrator of Mini Physics. } 16. \right] \end{equation}, \begin{equation} 109. \displaystyle{\frac{e^{ax}}{a^2-b^2} }[ -b \cosh bx + a \sinh bx ] & a\ne b \\ \int \frac{x}{\sqrt{a^2-x^2}}\ dx = -\sqrt{a^2-x^2} 28. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \sqrt{x} e^{ax}\ dx = \frac{1}{a}\sqrt{x}e^{ax} \int x \cos x \ dx = \cos x + x \sin x + (-1)^n \Gamma(n+1, ix)\right] \int \frac{1}{x}dx = \ln |x| Print on a single sheet side and obtaining the integrand on a single sheet side and obtaining integrand. 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