\right . \int e^{ax} \sinh bx \ dx = + \int \tan^2 ax\ dx = -x + \frac{1}{a} \tan ax \end{equation}, \begin{equation} Integral tables >> Basic forms. \int e^{-ax^2}\ dx = \frac{\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(x\sqrt{a}\right) Z xndx= xn+1 n+1 +C (n6= 1) 2. \int \cos^p ax dx = -\frac{1}{a(1+p)}{\cos^{1+p} ax} \times +\frac{1}{2}a^2\tan^{-1}\frac{x}{\sqrt{a^2-x^2}} -\frac{1}{a}{\cos ax} \hspace{2mm}{_2F_1}\left[ \text{ where erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt \end{equation}, \begin{equation} Notify me of follow-up comments by email. \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, $$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2}} = \sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+ \infty}_{- \infty}x^{2n} e^{-ax^{2}} = (-1)^{n} \frac{\partial^{n}}{\partial a^{n}}\sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2} + bx} = e^{\frac{b^2}{4a}}\sqrt{\frac{\pi}{a}}$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \sin^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 – \frac{6(-1)^n}{n^2 \pi^2} \right)$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \cos^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 + \frac{6(-1)^n}{n^2 \pi^2} \right)$$, $$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} xÂ \cos \left( \frac{ \pi x}{a} \right)Â \sin \left( \frac{2 \pi x}{a} \right) = \frac{8a^2}{9 \pi ^2}$$, $$\int\limits^{a}_{b} \frac{dx}{\sqrt{\left(a-x \right) \left(x-b \right)}} = \pi \text{ for a > b}$$, $$\int\limits^{a}_{b} \frac{dx}{x\sqrt{\left(a-x \right) \left(x-b \right)}} = \frac{ \pi}{\sqrt{ab}} \text{ for a > b > 0}$$, $$\int\limits^{\frac{\pi}{2}}_{- \frac{\pi}{2}} \frac{dx}{1+ y \sin x} = \frac{\pi}{\sqrt{1 – y^2}} \text{ for -1 < y < 1}$$, $$\int \frac{dx}{\sqrt{a^{2} – x^{2}}} = \text{arcsin} \, \frac{x}{a}$$, $$\int \frac{x dx}{\sqrt{a^{2} + x^{2}}} = \sqrt{a^{2} + x^{2}}$$, $$\int \frac{dx}{\sqrt{a^{2} +x^{2}}} = \text{ln} \, \left(x + \sqrt{a^{2} + x^{2}} \right)$$, $$\int \frac{dx}{a^{2} +x^{2}} = \frac{1}{a} \, \text{arctan} \, \frac{x}{a}$$, $$\int \frac{dx}{ \left( a^{2} + x^{2} \right)^{\frac{3}{2}}} = \frac{1}{a^{2}} \frac{x}{\sqrt{a^{2} +x^{2}}}$$, $$\int\frac{x \, dx}{ \left( a^{2}+x^{2} \right)^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{a^{2} + x^{2}}}$$, $$\int \frac{dx}{\sqrt{ (x – a)^{2} + b^{2}}} = \text{ln} \, \frac{1}{(a – x) + \sqrt{(a-x)^{2} + b^{2}}}$$, $$\int \frac{(x – a) \, dx}{\left[ (x-a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{(x-a)^{2} + b^{2}}}$$, $$\int \frac{dx}{\left[ (x – a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \frac{x – a}{b^{2} \sqrt{(x – a)^{2} +b^{2}}}$$. 34. \frac{b^2}{8a^2x}+ \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}} -\frac{\sin 2ax}{8a}- It is a compilation of the most commonly used integrals. Apr 30, 2018 - Complete table of integrals in a single sheet. \right] \int x\sqrt{x-a}\ dx = \end{equation}, \begin{equation} ÑÐ°Ð±Ð»Ð¸ÑÐ° Ð¸Ð½ÑÐµÐ³ÑÐ°Ð»Ð¾Ð². = uv " vdu ! \end{equation}, \begin{equation}\label{eq:Duley} Forms containing inverse trigonometric functions. \int \sin^n ax \ dx = \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)â1/4. 1. \end{split} It is essential for mathematicians, scientists, and engineers, who rely on it when identifying and subsequently solving extremely complex problems. \end{equation}, \begin{equation}\label{eq:Russ} \int x^n \cos x dx = \frac{1}{2}(ia)^{1-n}\left [ (-1)^n \Gamma(n+1, -iax) >> All the immediate integrals. \end{equation}, \begin{equation} \int x^2 \ln x \ dx = \frac{1}{3} x^3 \ln x-\frac{x^3}{9} 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 x e^x \cos x\ dx = \frac{1}{2}e^x (x \cos x \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \sinh ax \cosh ax dx= \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} 102. 4. -\frac{\sin[(2a-b)x]}{4(2a-b)} Integrals Involving a + bu, a â  0. \frac{1}{a^2 + b^2} \left[ \end{equation}, \begin{equation} \end{equation}, \begin{equation} The clustrmap is periodically (and automatically) archived and its counters reset, so the total is smaller. \end{equation}, \begin{equation} \int \cos ax \cosh bx\ dx = b \cos ax \cosh bx + \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:Gilmore} \int \frac{x}{a^2+x^2}dx = \frac{1}{2}\ln|a^2+x^2| \end{equation}, \begin{equation} \end{equation}, \begin{equation}\label{eq:ebke} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int x^2 \sin x\ dx = \left(2-x^2\right) \cos x + 2 x \sin x \int x(x+a)^n dx = \frac{(x+a)^{n+1} ( (n+1)x-a)}{(n+1)(n+2)} 112. + (-1)^n \Gamma(n+1, ix)\right] \\ & \left. \int e^{ax} \tanh bx\ dx = \int \sec^2 x \tan x\ dx = \frac{1}{2} \sec^2 x \text{ where } \Gamma(a,x)=\int_x^{\infty} t^{a-1}e^{-t}\hspace{2pt}\text{d}t \end{equation}, \begin{equation} \int \csc^nx \cot x\ dx = -\frac{1}{n}\csc^n x, n\ne 0 \int \ln ( x^2 + a^2 )\hspace{.5ex} {dx} = x \ln (x^2 + a^2 ) +2a\tan^{-1} \frac{x}{a} – 2x 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} \int \sqrt{x-a}\ dx = \frac{2}{3}(x-a)^{3/2} \int x \sqrt{x^2 \pm a^2}\ dx= \frac{1}{3}\left ( x^2 \pm a^2 \right)^{3/2} 111. \int \sqrt{\frac{x}{a+x}}\ dx = \sqrt{x(a+x)} \int x \ln \left ( a^2 – b^2 x^2 \right )\ dx = -\frac{1}{2}x^2+ \int \frac{x}{\sqrt{x\pm a} } \ dx = \frac{2}{3}(x\mp 2a)\sqrt{x\pm a} \int \sqrt{a x^2 + b x + c}\ dx = \end{equation}, \begin{equation} \end{equation}, \begin{equation} \end{equation}, \begin{equation} \int \frac{x}{ax^2+bx+c}dx = \frac{1}{2a}\ln|ax^2+bx+c| \frac{\sin[2(a-b)x]}{16(a-b)} \int \frac{1}{ax+b}dx = \frac{1}{a} \ln |ax + b| \int \frac{x^2}{\sqrt{x^2 \pm a^2}}\ dx = \frac{1}{2}x\sqrt{x^2 \pm a^2} 113. \end{equation}, \begin{equation} \right] \int \sec^2 ax\ dx = \frac{1}{a} \tan ax \end{equation}, \begin{equation} \int \sinh ax \cosh bx \ dx = \int \frac{x}{\sqrt{ax^2+bx+c}}\ dx= Free Integration Worksheet. \int x^2 e^{ax}\ dx = \left(\frac{x^2}{a}-\frac{2x}{a^2}+\frac{2}{a^3}\right) e^{ax} \int x \sin ax\ dx = -\frac{x \cos ax}{a} + \frac{\sin ax}{a^2} Table of Trig Integrals. \end{equation}, \begin{equation} 7. 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