Table Of Integrals

Rules for integration of general functions

$\backslash int\; cf(x)\backslash ,dx\; =\; c\backslash int\; f(x)\backslash ,dx$

$\backslash int+\; g(x)\backslash ,dx\; =\; \backslash int\; f(x)\backslash ,dx\; +\; \backslash int\; g(x)\backslash ,dx$

$\backslash int\; f(x)g(x)\backslash ,dx\; =\; f(x)\backslash int\; g(x)\backslash ,dx\; -\; \backslash int\; \backslash left(df(x)\backslash int\; g(x)\backslash ,dx\backslash right)$

Integrals of simple functions

Rational functions

more integrals: List of integrals of rational functions

$\backslash int\; \backslash ,dx\; =\; x\; +\; C$

$\backslash int\; x^n\backslash ,dx\; =\; \backslash frac\{x^\{n+1\}\}\{n+1\}\; +\; C\backslash qquad\backslash mbox\{\; if\; \}n\; \backslash ne\; -1$

$\backslash int\; \backslash frac\{1\}\{x\}\backslash ,dx\; =\; \backslash ln\{\backslash left|x\backslash right|\}\; +\; C$

$\backslash int\; \backslash frac\{1\}\{1+x^2\}\; \backslash ,\; dx\; =\; \backslash arctan\{x\}\; +\; C$

Logarithms

more integrals: List of integrals of logarithmic functions

$\backslash int\; \backslash ln\; \{x\}\backslash ,dx\; =\; x\; \backslash ln\; \{x\}\; -\; x\; +\; C$

$\backslash int\; \backslash log\_b\; \{x\}\backslash ,dx\; =\; x\backslash log\_b\; \{x\}\; -\; x\backslash log\_b\; \{e\}\; +\; C$

Exponential functions

more integrals: List of integrals of exponential functions

$\backslash int\; e^x\backslash ,dx\; =\; e^x\; +\; C$

$\backslash int\; a^x\backslash ,dx\; =\; \backslash frac\{a^x\}\{\backslash ln\{a\}\}\; +\; C$

Irrational functions

more integrals: List of integrals of irrational functions

$\backslash int\; \{1\; \backslash over\; \backslash sqrt\{1-x^2\}\}\; \backslash ,\; dx\; =\; \backslash arcsin\; \{x\}\; +\; C$

$\backslash int\; \{-1\; \backslash over\; \backslash sqrt\{1-x^2\}\}\; \backslash ,\; dx\; =\; \backslash arccos\; \{x\}\; +\; C$

$\backslash int\; \{x\; \backslash over\; \backslash sqrt\{x^2-1\}\}\; \backslash ,\; dx\; =\; \backslash mbox\{arcsec\}\backslash ,\{x\}\; +\; C$

Trigonometric functions

more integrals: List of integrals of trigonometric functions and List of integrals of arc functions

$\backslash int\; \backslash sin\{x\}\backslash ,\; dx\; =\; -\backslash cos\{x\}\; +\; C$

$\backslash int\; \backslash cos\{x\}\backslash ,\; dx\; =\; \backslash sin\{x\}\; +\; C$

$\backslash int\; \backslash tan\{x\}\; \backslash ,\; dx\; =\; -\backslash ln\{\backslash left|\; \backslash cos\; \{x\}\; \backslash right|\}\; +\; C$

$\backslash int\; \backslash cot\{x\}\; \backslash ,\; dx\; =\; \backslash ln\{\backslash left|\; \backslash sin\{x\}\; \backslash right|\}\; +\; C$

$\backslash int\; \backslash sec\{x\}\; \backslash ,\; dx\; =\; \backslash ln\{\backslash left|\; \backslash sec\{x\}\; +\; \backslash tan\{x\}\backslash right|\}\; +\; C$

$\backslash int\; \backslash csc\{x\}\; \backslash ,\; dx\; =\; -\backslash ln\{\backslash left|\; \backslash csc\{x\}\; +\; \backslash cot\{x\}\backslash right|\}\; +\; C$

$\backslash int\; \backslash sec^2\; x\; \backslash ,\; dx\; =\; \backslash tan\; x\; +\; C$

$\backslash int\; \backslash csc^2\; x\; \backslash ,\; dx\; =\; -\backslash cot\; x\; +\; C$

$\backslash int\; \backslash sec\{x\}\; \backslash ,\; \backslash tan\{x\}\; \backslash ,\; dx\; =\; \backslash sec\{x\}\; +\; C$

$\backslash int\; \backslash csc\{x\}\; \backslash ,\; \backslash cot\{x\}\; \backslash ,\; dx\; =\; -\; \backslash csc\{x\}\; +\; C$

$\backslash int\; \backslash sin^2\; x\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{2\}(x\; -\; \backslash sin\; x\; \backslash cos\; x)\; +\; C$

$\backslash int\; \backslash cos^2\; x\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{2\}(x\; +\; \backslash sin\; x\; \backslash cos\; x)\; +\; C$

$\backslash int\; \backslash sin^n\; x\; \backslash ,\; dx\; =\; -\; \backslash frac\{\backslash sin^\{n-1\}\; \{x\}\; \backslash cos\; \{x\}\}\{n\}\; +\; \backslash frac\{n-1\}\{n\}\; \backslash int\; \backslash sin^\{n-2\}\{x\}\; \backslash ,\; dx$

$\backslash int\; \backslash cos^n\; x\; \backslash ,\; dx\; =\; -\; \backslash frac\{\backslash cos^\{n-1\}\; \{x\}\; \backslash sin\; \{x\}\}\{n\}\; +\; \backslash frac\{n-1\}\{n\}\; \backslash int\; \backslash cos^\{n-2\}\{x\}\; \backslash ,\; dx$

$\backslash int\; \backslash arctan\{x\}\; \backslash ,\; dx\; =\; x\; \backslash ,\; \backslash arctan\{x\}\; +\; \backslash ln\; \backslash sqrt\{1\; +\; x^2\}\; +\; C$

Hyperbolic functions

more integrals: List of integrals of hyperbolic functions

$\backslash int\; \backslash sinh\; x\; \backslash ,\; dx\; =\; \backslash cosh\; x\; +\; C$

$\backslash int\; \backslash cosh\; x\; \backslash ,\; dx\; =\; \backslash sinh\; x\; +\; C$

$\backslash int\; \backslash tanh\; x\; \backslash ,\; dx\; =\; \backslash ln\; |\backslash cosh\; x|\; +\; C$

$\backslash int\; \backslash mbox\{csch\}\backslash ,x\; \backslash ,\; dx\; =\; \backslash ln\backslash left|\; \backslash tanh\; \{x\; \backslash over2\}\backslash right|\; +\; C$

$\backslash int\; \backslash mbox\{sech\}\backslash ,x\; \backslash ,\; dx\; =\; \backslash arctan(\backslash sinh\; x)\; +\; C$

$\backslash int\; \backslash coth\; x\; \backslash ,\; dx\; =\; \backslash ln|\backslash sinh\; x|\; +\; C$

Definite integrals

$\backslash int\_0^\backslash infty\{\backslash sqrt\{x\}\backslash ,e^\{-x\}\backslash ,dx\}\; =\; \backslash frac\{1\}\{2\}\backslash sqrt\; \backslash pi$

$\backslash int\_0^\backslash infty\{e^\{-x^2\}\backslash ,dx\}\; =\; \backslash frac\{1\}\{2\}\backslash sqrt\; \backslash pi$

$\backslash int\_0^\backslash infty\{\backslash frac\{x\}\{e^x-1\}\backslash ,dx\}\; =\; \backslash frac\{\backslash pi^2\}\{6\}$

$\backslash int\_0^\backslash infty\{\backslash frac\{x^3\}\{e^x-1\}\backslash ,dx\}\; =\; \backslash frac\{\backslash pi^4\}\{15\}$

$\backslash int\_0^\backslash infty\backslash frac\{\backslash sin(x)\}\{x\}\backslash ,dx=\backslash frac\{\backslash pi\}\{2\}$

$\backslash int\_0^\backslash infty\; x^\{z-1\}\backslash ,e^\{-x\}\backslash ,dx\; =\; \backslash Gamma(z)$

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