Integrals

$$\int x^n \, dx \quad (n \neq -1)$$

$$\frac{x^{n+1}}{n+1} + C$$

$$\int \frac{1}{x} \, dx$$

$$\ln(|x|) + C$$

$$\int e^x \, dx$$

$$e^x + C$$

$$\int a^x \, dx$$

$$\frac{a^x}{\ln(a)} + C$$

$$\int \ln(x) \, dx$$

$$x\ln(x) - x + C$$

$$\int \sin(x) \, dx$$

$$-\cos(x) + C$$

$$\int \cos(x) \, dx$$

$$\sin(x) + C$$

$$\int \sec^2(x) \, dx$$

$$\tan(x) + C$$

$$\int \csc^2(x) \, dx$$

$$-\cot(x) + C$$

$$\int \sec(x)\tan(x) \, dx$$

$$\sec(x) + C$$

$$\int \csc(x)\cot(x) \, dx$$

$$-\csc(x) + C$$

$$\int \tan(x) \, dx$$

$$-\ln(|\cos(x)|) + C = \ln(|\sec(x)|) + C$$

$$\int \cot(x) \, dx$$

$$\ln(|\sin(x)|) + C$$

$$\int \sec(x) \, dx$$

$$\ln(|\sec(x) + \tan(x)|) + C$$

$$\int \csc(x) \, dx$$

$$-\ln(|\csc(x) + \cot(x)|) + C$$

$$\int \sin^2(x) \, dx$$

$$\frac{x}{2} - \frac{\sin(2x)}{4} + C$$

$$\int \cos^2(x) \, dx$$

$$\frac{x}{2} + \frac{\sin(2x)}{4} + C$$

$$\int \frac{1}{\sqrt{1-x^2}} \, dx$$

$$\sin^{-1}(x) + C$$

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

$$\tan^{-1}(x) + C$$

$$\int \frac{1}{\sqrt{a^2-x^2}} \, dx$$

$$\sin^{-1}\left(\frac{x}{a}\right) + C$$

$$\int \frac{1}{a^2+x^2} \, dx$$

$$\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C$$

$$\int \frac{1}{|x|\sqrt{x^2-1}} \, dx$$

$$\sec^{-1}(x) + C$$

$$\int f(ax+b) \, dx$$

$\frac{1}{a}F(ax+b) + C$, where $F' = f$

$$\int \frac{f'(x)}{f(x)} \, dx$$

$$\ln(|f(x)|) + C$$

$\int u \, dv$ (Integration by parts)

$$uv - \int v \, du$$