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# Common integrals (primitive functions)

## Polynomials

$$\int dx = x + c \notag$$
$$\int k dx = kx + c \notag$$
$$\int x^2 dx = \frac {1}{n+1} x^{n+1} + c, n\neq -1 \notag$$
$$\int \frac {1}{x} dx = \int x^{-1} dx = \ln \left| x \right| + c \notag$$
$$\int x^{-n} dx = \frac {1}{1-n} x^{1-n} + c, n \neq 1 \notag$$
$$\int \frac {1}{ax+b} dx = \frac{1}{a} \ln \left| ax+b \right| + c \notag$$
$$\int x^{ \frac{p}{q} } dx = \frac {q}{p+q} x^{ \frac{p+q}{q}} +c \notag$$

## Trigonometric functions

$$\int \cos(x) dx = \sin(x) + c \notag$$
$$\int \sin(x) dx = -\cos(x) + c \notag$$
$$\int \sec^2(x) dx = \tan(x) + c \notag$$
$$\int \sec(x)\tan(x) dx = \sec(x) + c \notag$$
$$\int \csc(x)\cot(x) dx = -\csc(x) + c \notag$$
$$\int \csc^2(x) dx = -\cot(x) + c \notag$$
$$\int \tan(x) dx = \ln \left| sec(x) \right| + c \notag$$
$$\int \cot(x) dx = \ln \left| sin(x) \right| + c \notag$$
$$\int \sec(x) dx = \ln \left| \sec(x) + \tan(x) \right| + c \notag$$
$$\int \sec^3(x) dx = \frac{1}{2} (\sec(x)tan(x) + \ln \left| \sec(x) +\tan(x) \right|) + c \notag$$
$$\int \csc(x) dx = \ln \left| \csc(x) - \cot(x) \right| + c \notag$$
$$\int \csc^3(x) dx = \frac{1}{2} (\csc(x)cot(x) + \ln \left| \csc(x) - \cot(x) \right|) + c \notag$$

## Inverse trigonometric functions

$$\int \frac{1}{ \sqrt{a^2 - x^2} } dx = \sin^{-1} \left( \frac{x}{a} \right) + c\notag$$
$$\int \sin^{-1}(x) dx = x \sin^{-1}(x) + \sqrt{1-x^2} +c \notag$$
$$\int \frac{1}{ a^2 + x^2 } dx = \frac {1}{a} \tan^{-1} \left( \frac{x}{a} \right) + c\notag$$
$$\int \tan^{-1}(x) dx = x \tan^{-1}(x) - \frac{1}{2}\ln (1+x^2) +c \notag$$
$$\int \frac{1}{ x \sqrt{x^2 - a^2} } dx = \frac {1}{a} \sec^{-1} \left( \frac{x}{a} \right) + c\notag$$
$$\int \cos^{-1}(x) dx = x \cos^{-1}(x) - \sqrt{1-x^2} +c \notag$$

## Exponential and logarithm functions

$$\int \mathrm{e}^x dx = \mathrm{e}^x +c \notag$$
$$\int a^x dx = \frac {a^x}{ \ln(a)} + c \notag$$
$$\int \ln(x)dx = x\ln(x) - x + c \notag$$
$$\int x\mathrm{e}^x dx = (x-1)\mathrm{e}^x + c \notag$$
$$\int \frac{1}{x \ln(x) } dx = \ln \left| \ln(x) \right| + c \notag$$

## Hyperbolic trigonometric functions

$$\int \sinh(x)dx = \cosh(x) + c \notag$$
$$\int \cosh(x)dx = \sinh(x) + c \notag$$
$$\int \tanh(x)dx = \ln(\cosh(x)) + c \notag$$
$$\int \mathrm{sech}(x)\tanh(x) dx = -\mathrm{sech}(x) + c \notag$$
$$\int \mathrm{sech}^2(x) dx = \tanh(x) + c \notag$$
$$\int \mathrm{csch}(x)\coth(x) dx = -\mathrm{csch}(x) + c \notag$$
$$\int \mathrm{csch}^2(x) dx = - \coth(x) + c \notag$$
$$\int \mathrm{sech}(x) dx = \tan^{-1} \left| \sinh(x) \right| + c \notag$$