Integration

A list of common formulas, not in any way complete.
\[ \begin{align} \int 1 \dif x &= x + C & \int \frac{1}{x^2} \dif x &= -\frac{1}{x} + C \\ \int \frac{1}{\sqrt{x}} \dif x &= 2\sqrt{x} + C & \int \frac{1}{x} \dif x &= \ln \abs{x} + C \\ \int e^x \dif x &= e^x + C & \int a^x \dif x &= \frac{a^x}{\ln a} + C \end{align} \] \[ \begin{equation} \int x^n \dif x = \frac{x^{n+1}}{n + 1} + C \quad \text{with} \quad n \in \mathbb{R} \setminus \{-1\} \end{equation} \]
\[ \begin{align} \int \sin x \dif x &= -\cos x + C \\ \int \cos x \dif x &= \sin x + C \\ \int \frac{1}{\sin^2 x} \dif x &= -\cot x + C \\ \int \frac{1}{\cos^2 x} \dif x &= \tan x + C \\ \int \frac{1}{\sqrt{1 - x^2}} \dif x &= \arcsin x + C \\ &= -\arccos x + C \\ \int \frac{1}{1 + x^2} \dif x &= \arctan x + C \\ &= -\operatorname{arccot} x + C \end{align} \]
\[ \begin{align} \int f(x) + g(x) \dif x &= \int f(x) \dif x + \int g(x) \dif x \\ \int r \cdot f(x) \dif x &= r \int f(x) \dif x \end{align} \] \[ \begin{align} \int f(g(x)) \cdot g'(x) \dif x &= F(g(x)) + C \\ \int \frac{f'(x)}{f(x)} \dif x &= \ln \abs{f(x)} + C \end{align} \]

Trigonometric substitutions

Integration by parts