Basic Indefinite Integrals

Here is a list of commonly used basic indefinite integrals:

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + c \qquad \text{for } n \ne -1$$

$$\int x^{-1} \, dx = \int \frac{1}{x} \, dx = \ln |x| + c$$

$$\int \frac{1}{x} \, dx = \ln |x| + c$$

$$\int a^x \, dx = \frac{a^x}{\ln a} + c \qquad \text{for } a > 0,\, a \ne 1$$

$$\int e^x \, dx = e^x + c$$

$$\int \sin x \, dx = -\cos x + c$$

$$\int \cos x \, dx = \sin x + c$$

$$\int \frac{1}{\cos^2 x} \, dx = \tan x + c$$

$$\int \frac{1}{\sin^2 x} \, dx = -\cot x + c$$

$$\int \frac{1}{\sqrt{1 - x^2}} \, dx = \arcsin x + c$$

$$\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left( \frac{x}{a} \right) + c \qquad \text{for } a > 0$$

$$\int \frac{1}{1 + x^2} \, dx = \arctan x + c$$

$$\int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \cdot \arctan\left( \frac{x}{a} \right) + c \qquad \text{for } a > 0$$

From these formulas, we can also derive a number of generalized integrals involving composite functions:

$$\int f(x)^n \cdot f'(x) \, dx = \frac{f(x)^{n+1}}{n+1} + c \qquad \text{for } n \ne -1$$

$$\int \frac{f'(x)}{f(x)} \, dx = \ln |f(x)| + c$$

$$\int a^{f(x)} \cdot f'(x) \, dx = \frac{a^{f(x)}}{\ln a} + c \qquad \text{for } a > 0,\, a \ne 1$$

$$\int e^{f(x)} \cdot f'(x) \, dx = e^{f(x)} + c$$

$$\int \sin(f(x)) \cdot f'(x) \, dx = -\cos(f(x)) + c$$

$$\int \cos(f(x)) \cdot f'(x) \, dx = \sin(f(x)) + c$$

$$\int \frac{f'(x)}{\cos^2(f(x))} \, dx = \tan(f(x)) + c$$

$$\int \frac{f'(x)}{\sin^2(f(x))} \, dx = -\cot(f(x)) + c$$

$$\int \frac{f'(x)}{\sqrt{1 - f(x)^2}} \, dx = \arcsin(f(x)) + c$$

$$\int \frac{f'(x)}{\sqrt{a^2 - f(x)^2}} \, dx = \arcsin\left( \frac{f(x)}{a} \right) + c \qquad \text{for } a > 0$$

$$\int \frac{f'(x)}{1 + f(x)^2} \, dx = \arctan(f(x)) + c$$

And so on.