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Integration

Definition

Indefinite Intergral

\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \]
\[ \int dx = + C \]
\[ \int \frac{1}{x} dx = \ln|x|+ C \]
\[ \int e^x dx = e^x + C \]
\[ \int \sin(x) dx = - \cos(x) + C \]
\[ \int \cos(x) dx = \sin(x) + C \]
\[ \int \tan(x) dx = \ln|\sec(x)|+ C \]
\[ \int \cot(x) dx = -\ln|\csc(x)| + C \]
\[ \int \sec(x) dx = \ln|\sec(x) + \tan(x)|+ C \]
\[ \int \csc(x) dx = \ln|\csc(x) - \cot(x)| + C \]
\[ \int f(ax+b) dx = \frac{1}{a}f(ax+b) + C \]
\[ \int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + C \]

Integration by Parts

\[ \int u dv = uv - \int v du \]
\[ \int uv'dx = uv - \int vu'dx \]

Integration of Inverse Function

\[ \int f(x)dx = F(x) + C \]
\[ \int f^{-1}(x)dx = xf^{-1}(x) - F(f^{-1}(x)) + C \]

Integration by substitution

\[ \int f(x) dx = \int f(u(y)) u'(y) dy \]

Definite Integral

\[ \int_a^b f(x) dx = - \int_b^a f(x) dx \]
\[ \int_0^a f(x) dx = \int_0^a f(a-x) dx \]
\[ \int_a^b f(x) dx = \int_a^b f(a+b-x) dx \]
\[ \int_{-a}^{a} f(x) dx = { \begin{cases} 2 \int_0^a f(x) dx,&{\text{if }} f(x) = f(-x) \\ 0,&{\text{if }} f(x) \ne f(-x) \end{cases} } \]
\[ \int_a^b f(x) dx + \int_{f^{-1}(a)}^{f^{-1}(b)} f^{-1}(x) = bf^{-1}(b) - af^{-1}(a) \]