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

Integrals

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Hyperbolic Trig Functions

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

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

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

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

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

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

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

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

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

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

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

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

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

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:

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}dx=\sqrt{\pi }$$

$${\int }_0^{\mathrm{\infty }}{e}^{-x^2}dx=\frac{\sqrt{\pi }}{2}$$

Gamma Function:

$$\mathrm{\Gamma }(x)={\int }_0^{\mathrm{\infty }}{t}^{x-1}{e}^{-t}dt$$

$$\mathrm{\Gamma }(\frac{1}{2})={\int }_0^{\mathrm{\infty }}{t}^{-\frac{1}{2}}{e}^{-t}dt=\sqrt{\pi }$$

Beta Function:

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

$$B(z_1,z_2)={\int }_0^1{t}^{z_1-1}(1-t{)}^{z_2-1}dt$$

$$B(z_1,z_2)=2{\int }_0^{\frac{\pi }{2}}(\sin t{)}^{2z_1-1}(\cos t{)}^{2z_2-1}dt$$

$$B(z_1,z_2)=n{\int }_0^1{t}^{n{z}_1-1}(1-t^n{)}^{z_2-1}dt$$

$$B(z_1,z_2)={\int }_0^{\mathrm{\infty }}\frac{t^{z_1-1}}{(1+t{)}^{z_1+z_2}}dt$$

Power Reduction

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

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