Integrals - Hyperbolic Trig Functions - Special Functions - Gaussian Integral - Gamma Function - Beta Function - Power Reduction

Integrals

$${\int{\ln (x) dx} = x \ \ln(x) - x + C}$$

$${\int{\frac{1}{x} dx} = \ln(x) + C}$$

$${\int{\sin (x) dx} = -\cos(x) + C}$$

$${\int{\cos (x) dx} = \sin(x) + C}$$

$${\int{\sec (x) dx} = \ln |\sec(x) + \tan(x)| + C}$$

$${\int{\tan (x) dx} = \ln |\sec(x)| + C}$$

$${\int{\cot (x) dx} = \ln|\sin(x)| + C}$$

$${\int{\csc (x) dx} = \ln|\csc(x) - \cot(x)| + C}$$

$${\int{\csc (x) dx} = - \ln|\csc(x) + \cot(x)| + C}$$

$${\int{\sec^2 (x) dx} = \tan(x) + C}$$

$${\int{\csc^2 (x) dx} = -\cot(x) + C}$$

$${\int{\sec(x) \tan(x) dx} = \sec(x) + C}$$

$${\int{\sec^3 (x) dx} = \frac{1}{2} \sec(x) \tan(x) + \frac{1}{2} \ln |\sec(x) + \tan(x)| + C}$$

$${\int{\csc^3 (x) dx} = -\frac{1}{2} \csc(x) \cot(x) + \frac{1}{2} \ln |\csc(x) - \cot(x)| + C}$$

$${\int{\tan^2 (x) dx} = \tan(x) - x + C}$$

$${\int{\cot^2 (x) dx} = -\cot(x) - x + C}$$

$${\int{\frac{1}{a^2 + x^2} dx} = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C}$$

$${\int{\frac{1}{\sqrt{a^2 - x^2}} dx} = \sin^{-1}(\frac{x}{a}) + C}$$

$${\int{\frac{1}{x \sqrt{x^2 - a^2}} dx} = \frac{1}{a} \sec^{-1}|\frac{x}{a}| + C}$$

$${\int{\frac{1}{x^2 - a^2} dx} = -\frac{1}{a} \tanh^{-1}\left(\frac{x}{a}\right) + C}$$

$${\int{\frac{1}{x^2 - a^2} dx} = \frac{1}{2a} \ln \left\lvert \frac{x-a}{x+a} \right \rvert + C}$$

$${\int{\frac{1}{a^2 - x^2} dx} = \frac{1}{2a} \ln \left\lvert \frac{a+x}{a-x} \right \rvert + C}$$

$${\int{\frac{1}{\sqrt{1 + x^2}} dx} = \sinh^{-1}(x) + C}$$

$${\int{\frac{1}{\sqrt{x^2 - 1}} dx} = \cosh^{-1}(x) + C}$$

$${\int{\frac{1}{1 - x^2} dx} = \tanh^{-1}(x) + C}$$

Hyperbolic Trig Functions

$${\int{\sinh(x) dx} = \cosh(x) + C}$$

$${\int{\cosh(x) dx} = \sinh(x) + C}$$

$${\int{\tanh(x) dx} = \ln |\cosh(x)| + C}$$

$${\int{\coth(x) dx} = \ln|\sinh(x)| + C}$$

$${\int{\DeclareMathOperator{\sech}{sech}\sech(x) dx} = \tan^{-1}(\sinh(x)) + C}$$

$${\int{\DeclareMathOperator{\sech}{sech}\sech(x) dx} = 2 \ \tan^{-1}(e^x) + C}$$

$${\int{\DeclareMathOperator{\csch}{csch}\csch(x) dx} = \ln|\coth(x) - \DeclareMathOperator{\csch}{csch}\csch(x)| + C}$$

$${\int{\tanh^2(x) dx} = x - \tanh(x) + C}$$

$${\int{\coth^2(x) dx} = x - \coth(x) + C}$$

$${\int{\DeclareMathOperator{\sech}{sech}\sech^2(x) dx} = \tanh(x) + C}$$

$${\int{\DeclareMathOperator{\csch}{csch}\csch^2(x) dx} = - \coth(x) + C}$$

$${\int{\DeclareMathOperator{\sech}{sech}\sech(x) \tanh(x) dx} = -\DeclareMathOperator{\sech}{sech}\sech(x) + C}$$

$${\int{\DeclareMathOperator{\csch}{csch}\csch(x) \coth(x) dx} = -\DeclareMathOperator{\csch}{csch}\csch(x) + C}$$

Special Functions

$${\int e^{-x^2} dx = \frac{\sqrt{\pi}}{2} \text{erf}(x) + C}$$

$${\int e^{x^2} dx = \frac{\sqrt{\pi}}{2} \text{erfi}(x) + C}$$

$${\int \frac{e^{x}}{x} dx = \text{Ei}(x) + C}$$

$${\int \frac{1}{\ln(x)} dx = \text{li}(x) + C}$$

$${\int \frac{\sin(x)}{x} dx = \text{Si}(x) + C}$$

$${\int \frac{\cos(x)}{x} dx = \text{Ci}(x) + C}$$

$${\int \frac{\sin(x^2)}{x} dx = \text{S}(x) + C}$$

$${\int \frac{\cos(x^2)}{x} dx = \text{C}(x) + C}$$

Gaussian Integral:

$${\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}}$$

$${\int_{0}^{\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2}}$$

Gamma Function:

$${\Gamma(x) = \int_{0}^{\infty} t^{x-1} e^{-t} dt}$$

$${\Gamma(\frac{1}{2}) = \int_{0}^{\infty} t^{-\frac{1}{2}} e^{-t} dt = \sqrt{\pi}}$$

Beta Function:

$${B(z_1,z_2) = \frac{\Gamma(z_1)\Gamma(z_2)}{\Gamma(z_1 + z_2)}}$$

$${B(z_1,z_2) = \int_{0}^{1} t^{z_1 - 1} (1 - t)^{z_2 - 1} dt }$$

$${B(z_1,z_2) = 2 \int_{0}^{\frac{\pi}{2}} (\sin t)^{2z_1 - 1} (\cos t)^{2z_2 - 1} dt }$$

$${B(z_1,z_2) = n \int_{0}^{1} t^{n z_1 - 1} (1 - t^n)^{z_2 - 1} dt }$$

$${B(z_1,z_2) = \int_{0}^{\infty} \frac{t^{z_1 - 1}}{(1 + t)^{z_1 + z_2}} dt }$$

Power Reduction

$${\int \sin^n(x) dx = \frac{n - 1}{n} \int \sin^{n-2}(x) \ dx - \frac{\sin^{n-1}(x) \ \cos(x)}{n}}$$

$${\int \cos^n(x) dx = \frac{\cos^{n-1}(x) \ \sin(x)}{n} + \frac{n - 1}{n} \int \cos^{n-2}(x) \ dx}$$