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List of antiderivatives

Posted September 29th, 2009 by Structure
$$\int x^ndx=\frac{x^{n+1}}{n+1}+\Math{C}\;;\int f^n(x)f'(x)dx=\frac{f^{n+1}(x)}{n+1}+\Math{C}$$
$$\int \frac{1}{x}dx=\ln|x|+\Math{C}\;;\int \frac{f'(x)}{f(x)}dx=\ln|f(x)|+\Math{C}$$
$$\int \sin xdx=-\cos x+\Math{C}\;;\int \sin f(x)f'(x)dx=-\cos f(x)+\Math{C}$$
$$\int \frac{1}{\sin x}dx=\int \frac{1}{2\sin \frac{x}{2}\cos\frac{x}{2}}dx=\int \frac{1}{2\tan\frac{x}{2}\cos^2\frac{x}{2}}dx=\ln|\tan\frac{x}{2}|+\Math{C}$$
$$\int \cos xdx=\sin x +\Math{C}\;;\int \cos f(x)f'(x)dx=\sin f(x)+\Math{C}$$
$$\int \frac{1}{\cos x}dx=\int \frac{\cos x}{\cos^2 x}=\int \frac{\cos x}{1-\sin^2 x}dx=\frac{1}{2}\ln|\frac{1+\sin x}{1-\sin x}|+\math{C}=\frac{1}{2}\ln|\frac{1+2\sin \frac{x}{2}\cos\frac{x}{2}}{1-2\sin \frac{x}{2}\cos\frac{x}{2}}|+\Math{C}=\ln|\frac{\sin \frac{x}{2}+\cos\frac{x}{2}}{\sin \frac{x}{2}-\cos\frac{x}{2}}|+\Math{C}$$
$$\int \sin axdx=-\frac{1}{a}\cos ax+\Math{C}$$
$$\int \cos axdx=\frac{1}{a}\sin ax +\Math{C}$$
$$\int e^xdx=e^x+\Math{C}\;;\int e^{f(x)}f'(x)dx=e^{f(x)}+\Math{C}$$
$$\int e^{ax}dx=\frac{1}{a}e^{ax}+\Math{C}$$
$$\int \frac{1}{\sqrt{x^2+1}}dx=\ln(x+\sqrt{x^2+1})+\Math{C}$$
$$\int \frac{1}{x^2+1}dx=\arctan x+\Math{C}$$
$$\int \frac{1}{x^2+a^2}dx=\frac{1}{a}\arctan \frac{x}{a}+\Math{C}$$
$$\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}\ln|\frac{x-a}{x+a}|+\Math{C}$$
$$\int \frac{1}{\sqrt{1-x^2}}dx=\arcsin x+\Math{C}$$
$$\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin \frac{x}{a}+\Math{C}$$
$$\int \frac{x}{\sqrt{1-x^2}}dx=-\sqrt{1-x^2}+\Math{C}$$
$$\int \frac{x}{\sqrt{1+x^2}}dx=\sqrt{1+x^2}+\Math{C}$$
$$\int \sqrt{x^2+a^2}dx=\frac{1}{2}x\sqrt{x^2+a^2}+\frac{a^2}{2}\ln(x+\sqrt{x^2+a^2})+\math{C}$$
$$\int f(x)g'(x) dx=f(x)g(x)-\int f'(x)g(x)dx$$
$$\int \ln x dx=x\ln x-x+\Math{C}$$
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