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