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Integral Table


Basic Forms

$$\int x^{n} d x=\frac{1}{n+1} x^{n+1}, \quad n \neq-1$$ $$\int \frac{1}{x} d x=\ln |x|$$ $$\int u d v=u v-\int v d u$$ $$\int \frac{1}{a x+b} d x=\frac{1}{a} \ln |a x+b|$$

Integrals of Rational Functions

$$\int \frac{1}{(x+a)^{2}} d x=-\frac{1}{x+a}$$ $$\int(x+a)^{n} d x=\frac{(x+a)^{n+1}}{n+1}, n \neq-1$$ $$\int x(x+a)^{n} d x=\frac{(x+a)^{n+1}((n+1) x-a)}{(n+1)(n+2)}$$ $$\int \frac{1}{1+x^{2}} d x=\tan ^{-1} x$$ $$\int \frac{1}{a^{2}+x^{2}} d x=\frac{1}{a} \tan ^{-1} \frac{x}{a}$$ $$\int \frac{x}{a^{2}+x^{2}} d x=\frac{1}{2} \ln \left|a^{2}+x^{2}\right|$$ $$\int \frac{x^{2}}{a^{2}+x^{2}} d x=x-a \tan ^{-1} \frac{x}{a}$$ $$\int \frac{x^{3}}{a^{2}+x^{2}} d x=\frac{1}{2} x^{2}-\frac{1}{2} a^{2} \ln \left|a^{2}+x^{2}\right|$$ $$\int \frac{1}{a x^{2}+b x+c} d x=\frac{2}{\sqrt{4 a c-b^{2}}} \tan ^{-1} \frac{2 a x+b}{\sqrt{4 a c-b^{2}}}$$ $$\int \frac{1}{(x+a)(x+b)} d x=\frac{1}{b-a} \ln \frac{a+x}{b+x}, a \neq b$$ $$\int \frac{x}{(x+a)^{2}} d x=\frac{a}{a+x}+\ln |a+x|$$ $$\int \frac{x}{a x^{2}+b x+c} d x=\frac{1}{2 a} \ln \left|a x^{2}+b x+c\right|-\frac{b}{a \sqrt{4 a c-b^{2}}} \tan ^{-1} \frac{2 a x+b}{\sqrt{4 a c-b^{2}}}$$

Integrals with Roots

$$\int \sqrt{x-a} \quad d x=\frac{2}{3}(x-a)^{3 / 2}$$ $$\int \frac{1}{\sqrt{x \pm a}} \quad d x=2 \sqrt{x \pm a}$$ $$\int \frac{1}{\sqrt{a-x}} \quad d x=-2 \sqrt{a-x}$$ $$\int x \sqrt{x-a} \quad d x=\left\{\begin{array}{l}\frac{2 a}{3}(x-a)^{3 / 2}+\frac{2}{5}(x-a)^{5 / 2}, \text { or } \\ \frac{2}{3} x(x-a)^{3 / 2}-\frac{4}{15}(x-a)^{5 / 2}, \text { or } \\ \frac{2}{15}(2 a+3 x)(x-a)^{3 / 2}\end{array}\right.$$ $$\int \sqrt{a x+b} \quad d x=\left(\frac{2 b}{3 a}+\frac{2 x}{3}\right) \sqrt{a x+b}$$ $$\int(a x+b)^{3 / 2} \quad d x=\frac{2}{5 a}(a x+b)^{5 / 2}$$ $$\int \frac{x}{\sqrt{x \pm a}} \quad d x=\frac{2}{3}(x \mp 2 a) \sqrt{x \pm a}$$ $$\int \sqrt{\frac{x}{a-x}} \quad d x=-\sqrt{x(a-x)}-a \tan ^{-1} \frac{\sqrt{x(a-x)}}{x-a}$$ $$\int \sqrt{\frac{x}{a+x}} \quad d x=\sqrt{x(a+x)}-a \ln [\sqrt{x}+\sqrt{x+a}]$$ $$\int x \sqrt{a x+b} \quad d x=\frac{2}{15 a^{2}}\left(-2 b^{2}+a b x+3 a^{2} x^{2}\right) \sqrt{a x+b}$$ $$\int \sqrt{x(a x+b)} \quad d x=\frac{1}{4 a^{3 / 2}}\left[(2 a x+b) \sqrt{a x(a x+b)}-b^{2} \ln |a \sqrt{x}+\sqrt{a(a x+b)}|\right]$$ $$\int \sqrt{x^{3}(a x+b)} \quad d x=\left[\frac{b}{12 a}-\frac{b^{2}}{8 a^{2} x}+\frac{x}{3}\right] \sqrt{x^{3}(a x+b)}+\frac{b^{3}}{8 a^{5 / 2}} \ln |a \sqrt{x}+\sqrt{a(a x+b)}|$$ $$\int \sqrt{x^{2} \pm a^{2}} \quad d x=\frac{1}{2} x \sqrt{x^{2} \pm a^{2}} \pm \frac{1}{2} a^{2} \ln |x+\sqrt{x^{2} \pm a^{2}}|$$ $$\int \sqrt{a^{2}-x^{2}} \quad d x=\frac{1}{2} x \sqrt{a^{2}-x^{2}}+\frac{1}{2} a^{2} \tan ^{-1} \frac{x}{\sqrt{a^{2}-x^{2}}}$$ $$\int x \sqrt{x^{2} \pm a^{2}} \quad d x=\frac{1}{3}\left(x^{2} \pm a^{2}\right)^{3 / 2}$$ $$\int \frac{1}{\sqrt{x^{2} \pm a^{2}}} \quad d x=\ln |x+\sqrt{x^{2} \pm a^{2}}|$$ $$\int \frac{1}{\sqrt{a^{2}-x^{2}}} \quad d x=\sin ^{-1} \frac{x}{a}$$ $$\int \frac{x}{\sqrt{x^{2} \pm a^{2}}} \quad d x=\sqrt{x^{2} \pm a^{2}}$$ $$\int \frac{x}{\sqrt{a^{2}-x^{2}}} \quad d x=-\sqrt{a^{2}-x^{2}}$$ $$\int \frac{x^{2}}{\sqrt{x^{2} \pm a^{2}}} \quad d x=\frac{1}{2} x \sqrt{x^{2} \pm a^{2}} \mp \frac{1}{2} a^{2} \ln |x+\sqrt{x^{2} \pm a^{2}}|$$ $$\int \sqrt{a x^{2}+b x+c} \quad d x=\frac{b+2 a x}{4 a} \sqrt{a x^{2}+b x+c}+\frac{4 a c-b^{2}}{8 a^{3 / 2}} \ln |2 a x+b+2 \sqrt{a\left(a x^{2}+b x+ c\right)}|$$ $$\int x \sqrt{a x^{2}+b x+c} \quad d x=\frac{1}{48 a^{5 / 2}}\left(2 \sqrt{a} \sqrt{a x^{2}+b x+c}\left(-3 b^{2}+2 a b x+8 a\left(c+a x^{2}\right)\right)\right.$$ $$\left.+3\left(b^{3}-4 a b c\right) \ln |b+2 a x+2 \sqrt{a} \sqrt{a x^{2}+b x+c}|\right)$$ $$\int \frac{1}{\sqrt{a x^{2}+b x+c}} \quad d x=\frac{1}{\sqrt{a}} \ln |2 a x+b+2 \sqrt{a\left(a x^{2}+b x+c\right)}|$$ $$\int \frac{x}{\sqrt{a x^{2}+b x+c}} d x=\frac{1}{a} \sqrt{a x^{2}+b x+c}-\frac{b}{2 a^{3 / 2}} \ln |2 a x+b+2 \sqrt{a\left(a x^{2}+b x+c\right)}|$$ $$\int \frac{d x}{\left(a^{2}+x^{2}\right)^{3 / 2}}=\frac{x}{a^{2} \sqrt{a^{2}+x^{2}}}$$

Integrals with Logarithms

$$\int \ln a x \quad d x=x \ln a x-x$$ $$\int x \ln x \quad d x=\frac{1}{2} x^{2} \ln x-\frac{x^{2}}{4}$$ $$\int x^{2} \ln x \quad d x=\frac{1}{3} x^{3} \ln x-\frac{x^{3}}{9}$$ $$\int x^{n} \ln x \quad d x=x^{n+1}\left(\frac{\ln x}{n+1}-\frac{1}{(n+1)^{2}}\right), \quad n \neq-1$$ $$\int \frac{\ln a x}{x} \quad d x=\frac{1}{2}(\ln a x)^{2}$$ $$\int \frac{\ln x}{x^{2}} \quad d x=-\frac{1}{x}-\frac{\ln x}{x}$$ $$\int \ln (a x+b) \quad d x=\left(x+\frac{b}{a}\right) \ln (a x+b)-x, a \neq 0$$ $$\int \ln \left(x^{2}+a^{2}\right) d x=x \ln \left(x^{2}+a^{2}\right)+2 a \tan ^{-1} \frac{x}{a}-2 x$$ $$\int \ln \left(x^{2}-a^{2}\right) d x=x \ln \left(x^{2}-a^{2}\right)+a \ln \frac{x+a}{x-a}-2 x$$ $$\int \ln \left(a x^{2}+b x+c\right) \quad d x=\frac{1}{a} \sqrt{4 a c-b^{2}} \tan ^{-1} \frac{2 a x+b}{\sqrt{4 a c-b^{2}}}-2 x+\left(\frac{b}{2 a}+x\right) \ln \left(a x^{2}+b x+c\right)$$ $$\int x \ln (a x+b) \quad d x=\frac{b x}{2 a}-\frac{1}{4} x^{2}+\frac{1}{2}\left(x^{2}-\frac{b^{2}}{a^{2}}\right) \ln (a x+b)$$ $$\int x \ln \left(a^{2}-b^{2} x^{2}\right) \quad d x=-\frac{1}{2} x^{2}+\frac{1}{2}\left(x^{2}-\frac{a^{2}}{b^{2}}\right) \ln \left(a^{2}-b^{2} x^{2}\right)$$ $$\int(\ln x)^{2} \quad d x=2 x-2 x \ln x+x(\ln x)^{2}$$ $$\int(\ln x)^{3} \quad d x=-6 x+x(\ln x)^{3}-3 x(\ln x)^{2}+6 x \ln x$$ $$\int x(\ln x)^{2} \quad d x=\frac{x^{2}}{4}+\frac{1}{2} x^{2}(\ln x)^{2}-\frac{1}{2} x^{2} \ln x$$ $$\int x^{2}(\ln x)^{2} \quad d x=\frac{2 x^{3}}{27}+\frac{1}{3} x^{3}(\ln x)^{2}-\frac{2}{9} x^{3} \ln x$$

Integrals with Exponentials

$$\int e^{a x} \quad d x=\frac{1}{a} e^{a x}$$ $$\int \sqrt{x} e^{a x} \quad d x=\frac{1}{a} \sqrt{x} e^{a x}+\frac{i \sqrt{\pi}}{2 a^{3 / 2}} \operatorname{erf}(i \sqrt{a x}),\text{ where erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t$$ $$\int x e^{x} \quad d x=(x-1) e^{x}$$ $$\int x e^{a x} \quad d x=\left(\frac{x}{a}-\frac{1}{a^{2}}\right) e^{a x}$$ $$\int x^{2} e^{x} \quad d x=\left(x^{2}-2 x+2\right) e^{x}$$ $$\int x^{2} e^{a x} \quad d x=\left(\frac{x^{2}}{a}-\frac{2 x}{a^{2}}+\frac{2}{a^{3}}\right) e^{a x}$$ $$\int x^{3} e^{x} \quad d x=\left(x^{3}-3 x^{2}+6 x-6\right) e^{x}$$ $$\int x^{n} e^{a x} \quad d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} \mathrm{d} x$$ $$ \int x^{n} e^{a x} \quad d x=\frac{(-1)^{n}}{a^{n+1}} \Gamma[1+n,-a x], \text { where } \Gamma(a, x)=\int_{x}^{\infty} t^{a-1} e^{-t} \mathrm{d} t $$ $$\int e^{a x^{2}} \quad d x=-\frac{i \sqrt{\pi}}{2 \sqrt{a}} \operatorname{erf}(i x \sqrt{a})$$ $$\int e^{-a x^{2}} \quad d x=\frac{\sqrt{\pi}}{2 \sqrt{a}} \operatorname{erf}(x \sqrt{a})$$ $$\int x e^{-a x^{2}} \quad d x=-\frac{1}{2 a} e^{-a x^{2}}$$ $$\int x^{2} e^{-a x^{2}} \quad d x=\frac{1}{4} \sqrt{\frac{\pi}{a^{3}}} \operatorname{erf}(x \sqrt{a})-\frac{x}{2 a} e^{-a x^{2}}$$

Integrals with Trigonometric Functions

$$\int \sin a x \quad d x=-\frac{1}{a} \cos a x$$ $$\int \sin ^{2} a x \quad d x=\frac{x}{2}-\frac{\sin 2 a x}{4 a}$$ $$\int \sin ^{3} a x \quad d x=-\frac{3 \cos a x}{4 a}+\frac{\cos 3 a x}{12 a}$$ $$\int \sin ^{n} a x \quad d x=-\frac{1}{a} \cos a x \quad_{2} F_{1}\left[\frac{1}{2}, \frac{1-n}{2}, \frac{3}{2}, \cos ^{2} a x\right]$$ $$\int \cos a x \quad d x=\frac{1}{a} \sin a x$$ $$\int \cos ^{2} a x \quad d x=\frac{x}{2}+\frac{\sin 2 a x}{4 a}$$ $$\int \cos ^{3} a x d x=\frac{3 \sin a x}{4 a}+\frac{\sin 3 a x}{12 a}$$ $$\int \cos ^{p} a x d x=-\frac{1}{a(1+p)} \cos ^{1+p} a x \times_{2} F_{1}\left[\frac{1+p}{2}, \frac{1}{2}, \frac{3+p}{2}, \cos ^{2} a x\right]$$ $$\int \cos x \sin x \quad d x=\frac{1}{2} \sin ^{2} x+c_{1}=-\frac{1}{2} \cos ^{2} x+c_{2}=-\frac{1}{4} \cos 2 x+c_{3}$$ $$\int \cos a x \sin b x \quad d x=\frac{\cos [(a-b) x]}{2(a-b)}-\frac{\cos [(a+b) x]}{2(a+b)}, a \neq b$$ $$\int \sin ^{2} a x \cos b x \quad d x=-\frac{\sin [(2 a-b) x]}{4(2 a-b)}+\frac{\sin b x}{2 b}-\frac{\sin [(2 a+b) x]}{4(2 a+b)}$$ $$\int \sin ^{2} x \cos x \quad d x=\frac{1}{3} \sin ^{3} x$$ $$\int \cos ^{2} a x \sin b x \quad d x=\frac{\cos [(2 a-b) x]}{4(2 a-b)}-\frac{\cos b x}{2 b}-\frac{\cos [(2 a+b) x]}{4(2 a+b)}$$ $$\int \cos ^{2} a x \sin a x \quad d x=-\frac{1}{3 a} \cos ^{3} a x$$ $$\int \sin ^{2} a x \cos ^{2} b x d x=\frac{x}{4}-\frac{\sin 2 a x}{8 a}-\frac{\sin [2(a-b) x]}{16(a-b)}+\frac{\sin 2 b x}{8 b}-\frac{\sin [2(a+b) x]}{16(a+b)}$$ $$\int \sin ^{2} a x \cos ^{2} a x \quad d x=\frac{x}{8}-\frac{\sin 4 a x}{32 a}$$ $$\int \tan a x \quad d x=-\frac{1}{a} \ln \cos a x$$ $$\int \tan ^{2} a x \quad d x=-x+\frac{1}{a} \tan a x$$ $$\int \tan ^{n} a x \quad d x=\frac{\tan ^{n+1} a x}{a(1+n)} \times_{2} F_{1}\left(\frac{n+1}{2}, 1, \frac{n+3}{2},-\tan ^{2} a x\right)$$ $$\int \tan ^{3} a x d x=\frac{1}{a} \ln \cos a x+\frac{1}{2 a} \sec ^{2} a x$$ $$\int \sec x d x=\ln |\sec x+\tan x|=2 \tanh ^{-1}\left(\tan \frac{x}{2}\right)$$ $$\int \sec ^{2} a x d x=\frac{1}{a} \tan a x$$ $$\int \sec ^{3} x d x=\frac{1}{2} \sec x \tan x+\frac{1}{2} \ln |\sec x+\tan x|$$ $$\int \sec x \tan x \quad d x=\sec x$$ $$\int \sec ^{2} x \tan x \quad d x=\frac{1}{2} \sec ^{2} x$$ $$\int \sec ^{n} x \tan x \quad d x=\frac{1}{n} \sec ^{n} x, n \neq 0$$ $$\int \csc x \quad d x=\ln \left|\tan \frac{x}{2}\right|=\ln |\csc x-\cot x|$$ $$\int \csc ^{2} a x \quad d x=-\frac{1}{a} \cot a x$$ $$\int \csc ^{3} x \quad d x=-\frac{1}{2} \cot x \csc x+\frac{1}{2} \ln |\csc x-\cot x|$$ $$\int \csc ^{n} x \cot x \quad d x=-\frac{1}{n} \csc ^{n} x, n \neq 0$$ $$\int \sec x \csc x \quad d x=\ln |\tan x|$$

Products of Trigonometric Functions and Monomials

$$\int x \cos x \quad d x=\cos x+x \sin x$$ $$\int x \cos a x \quad d x=\frac{1}{a^{2}} \cos a x+\frac{x}{a} \sin a x$$ $$\int x^{2} \cos x \quad d x=2 x \cos x+\left(x^{2}-2\right) \sin x$$ $$\int x^{2} \cos a x \quad d x=\frac{2 x \cos a x}{a^{2}}+\frac{a^{2} x^{2}-2}{a^{3}} \sin a x$$ $$\int x^{n} \cos x d x=-\frac{1}{2}(i)^{n+1}\left[\Gamma(n+1,-i x)+(-1)^{n} \Gamma(n+1, i x)\right]$$ $$\int x^{n} \cos a x \quad d x=\frac{1}{2}(i a)^{1-n}\left[(-1)^{n} \Gamma(n+1,-i a x)-\Gamma(n+1, i x a)\right]$$ $$\int x \sin x \quad d x=-x \cos x+\sin x$$ $$\int x \sin a x \quad d x=-\frac{x \cos a x}{a}+\frac{\sin a x}{a^{2}}$$ $$\int x^{2} \sin x \quad d x=\left(2-x^{2}\right) \cos x+2 x \sin x$$ $$\int x^{2} \sin a x \quad d x=\frac{2-a^{2} x^{2}}{a^{3}} \cos a x+\frac{2 x \sin a x}{a^{2}}$$ $$\int x^{n} \sin x \quad d x=-\frac{1}{2}(i)^{n}\left[\Gamma(n+1,-i x)-(-1)^{n} \Gamma(n+1,-i x)\right]$$ $$\int x \cos ^{2} x \quad d x=\frac{x^{2}}{4}+\frac{1}{8} \cos 2 x+\frac{1}{4} x \sin 2 x$$ $$\int x \sin ^{2} x d x=\frac{x^{2}}{4}-\frac{1}{8} \cos 2 x-\frac{1}{4} x \sin 2 x$$ $$\int x \tan ^{2} x d x=-\frac{x^{2}}{2}+\ln \cos x+x \tan x$$ $$\int x \sec ^{2} x \quad d x=\ln \cos x+x \tan x$$

Products of Trigonometric Functions and Exponentials

$$\int e^{x} \sin x \quad d x=\frac{1}{2} e^{x}(\sin x-\cos x)$$ $$\int e^{b x} \sin a x \quad d x=\frac{1}{a^{2}+b^{2}} e^{b x}(b \sin a x-a \cos a x)$$ $$\int e^{x} \cos x \quad d x=\frac{1}{2} e^{x}(\sin x+\cos x)$$ $$\int e^{b x} \cos a x \quad d x=\frac{1}{a^{2}+b^{2}} e^{b x}(a \sin a x+b \cos a x)$$ $$\int x e^{x} \sin x \quad d x=\frac{1}{2} e^{x}(\cos x-x \cos x+x \sin x)$$ $$\int x e^{x} \cos x \quad d x=\frac{1}{2} e^{x}(x \cos x-\sin x+x \sin x)$$

Integrals of Hyperbolic Functions

$$\int \cosh a x \quad d x=\frac{1}{a} \sinh a x$$ $$\int e^{a x} \cosh b x \quad d x=\left\{\begin{array}{ll}\frac{e^{a x}}{a^{2}-b^{2}}[a \cosh b x-b \sinh b x] & a \neq b \\ \frac{e^{2 a x}}{4 a}+\frac{x}{2} & a=b\end{array}\right.$$ $$\int \sinh a x \quad d x=\frac{1}{a} \cosh a x$$ $$\int e^{a x} \sinh b x \quad d x=\left\{\begin{array}{ll}\frac{e^{a x}}{a^{2}-b^{2}}[-b \cosh b x+a \sinh b x] & a \neq b \\ \frac{e^{2 a x}}{4 a}-\frac{x}{2} & a=b\end{array}\right.$$ $$\int e^{a x} \tanh b x \quad d x=\left\{\begin{array}{ll}\frac{e^{(a+2 b) x}}{(a+2 b)}\left(\quad {}_2 F_{1}\right)\left[1+\frac{a}{2 b}, 1,2+\frac{a}{2 b},-e^{2 b x}\right] \\ -\frac{e^{a x}}{a}\left(\quad {}_2 F_{1}\right)\left[1, \frac{a}{2 b}, 1+\frac{a}{2 b},-e^{2 b x}\right] \\ \frac{e^{ax}-2 \tan ^{-1}\left[e^{a x}\right]}{a}\end{array}\right.$$ $$\int \cos a x \cosh b x \quad d x=\frac{1}{a^{2}+b^{2}}[a \sin a x \cosh b x+b \cos a x \sinh b x]$$ $$\int \cos a x \sinh b x \quad d x=\frac{1}{a^{2}+b^{2}}[b \cos a x \cosh b x+a \sin a x \sinh b x]$$ $$\int \sin a x \cosh b x \quad d x=\frac{1}{a^{2}+b^{2}}[-a \cos a x \cosh b x+b \sin a x \sinh b x]$$ $$\int \sin a x \sinh b x \quad d x=\frac{1}{a^{2}+b^{2}}[b \cosh b x \sin a x-a \cos a x \sinh b x]$$ $$\int \sinh a x \cosh a x d x=\frac{1}{4 a}[-2 a x+\sinh 2 a x]$$ $$\int \sinh a x \cosh b x \quad d x=\frac{1}{b^{2}-a^{2}}[b \cosh b x \sinh a x-a \cosh a x \sinh b x]$$

Linearity

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