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Math Formulas: Common Integrals

Indefinite Integral

Method of substitution

$$ \int f\left(g(x)\right)\cdot g'(x) dx = \int f(u) du $$

Integration by parts

$$ \int f(x) \cdot g'(x)dx = f(x) \cdot g(x) - \int g(x) \cdot f'(x)dx $$

Integrals of Rational and Irrational Functions

$$ \int x^n dx = \frac{x^{n+1}}{n+1} + C , n \ne 1 $$
$$ \int \frac{1}{x} dx = \ln|x| + C $$
$$ \int c \, dx = c \cdot x + C $$
$$ \int x \, dx = \frac{x^2}{2} + C $$
$$ \int x^2 \, dx = \frac{x^3}{3} + C $$
$$ \int \frac{1}{x^2} dx = -\frac{1}{x} + C $$
$$ \int \sqrt{x} \, dx = \frac{2\cdot x \cdot \sqrt{x} }{3} + C $$
$$ \int \frac{1}{1+x^2} dx = \arctan x + C $$
$$ \int \frac{1}{\sqrt{1-x^2}} dx = \arcsin x + C $$

Integrals of Trigonometric Functions

$$ \int \sin x\,dx = -\cos x + C $$
$$ \int \cos x\,dx = \sin x + C $$
$$ \int \tan x\,dx = \ln|\sec x| + C $$
$$ \int \sec x\,dx = \ln|\tan x + \sec x | + C $$
$$ \int \sin^2x\,dx = \frac{1}{2}(x-\sin x \cdot \cos x) + C $$
$$ \int \cos^2x\,dx = \frac{1}{2}(x + \sin x \cdot \cos x) + C $$
$$ \int \tan^2x\,dx = \tan x - x + C $$
$$ \int \sec^2x\,dx = \tan x + C $$

Integrals of Exponential and Logarithmic Functions

$$ \int \ln x \,dx =x \cdot \ln x -x + C $$
$$ \int x^n \cdot \ln x \,dx =\frac{x^{n+1}}{n+1} \ln x - \frac{x^{n+1}}{(n+1)^2} + C $$
$$ \int e^x\,dx = e^x + C $$
$$ \int a^x\,dx = \frac{a^x}{\ln a} + C $$

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