# Integration formulas

It is considered a and \text{C} as real or constant numbers, u is the function of x and u' is the derivative of u.

## Basic forms and properties of integrals

• \displaystyle \int \ u^{n} \cdot du = \cfrac{u^{(n+1)}}{n+1} + \text{C},\qquad n \neq -1
• \displaystyle \int \ du = u + \text{C}
• \displaystyle \int a \cdot (u^{n}) \cdot du = \displaystyle a \cdot \int u^{n} \cdot du = a \cdot \Bigg ( \cfrac{u^{(n + 1)}}{n + 1} \Bigg ) + \text{C}, \qquad[katex] n \neq -1[/katex]
• \displaystyle \int \cfrac{du}{u} = \ln |u| + \text{C}
• \displaystyle \int e^{u} \cdot du = e^{u} + \text{C}
• \displaystyle \int a^{u} \cdot du = \cfrac{a^{u}}{\ln a} + \text{C}

#### The integral of an addition / subtraction is equal to the addition / subtraction of the integrals

• \displaystyle \int \big (u^{n} \pm u^{n\pm 1} \pm u^{n\pm 2} \big ) \cdot du = \displaystyle \int u^{n} \cdot du \pm \int u^{n \pm 1} \cdot du \pm \int u^{n\pm 2} \cdot du

#### Integral of sine

• \displaystyle \int \sin u \cdot du = - \cos u + \text{C}

#### Integral of cosine

• \displaystyle \int \cos u \cdot du = \sin u + \text{C}

#### Integral of tangent

• \displaystyle \int \tan u \cdot du = \ln |\sec u| + \text{C} = -\ln |\cos u| + \text{C}

#### Integral of cotangent

• \displaystyle \int \cot u \cdot du = \ln |\sin u|+ \text{C}

#### Integral of secant

• \displaystyle \int \sec u \cdot du = \log\left(\sec u + \tan u\right)+ \text{C}

#### Integral of cosecant

• \displaystyle \int \csc u \cdot du = \ln |\csc u - \cot u|+ \text{C} =   -\csc u \cdot \cot u + \text{C} = -\log\left( \cot u + \csc u\right)

#### Integral of secant by tangent

• \displaystyle \int \sec u \cdot \tan u \cdot du = \sec u + \text{C}

#### Integral of cosecant by cotangent

• \displaystyle \int \csc u \cdot \cot u \cdot du = - \csc u + \text{C}

## Several integrals

• \displaystyle \int \cfrac{du}{u^{2} + a^{2}} = \cfrac{1}{a} \cdot \tan^{-1} \cfrac{u}{a} + \text{C}
• \displaystyle \int \cfrac{du}{u^{2} - a^{2}} = \cfrac{1}{2 \cdot a} \cdot \ln \bigg | \cfrac{u - a}{u + a} \bigg | + \text{C}
• \displaystyle \int \cfrac{du}{a^{2} - u^{2}} = \cfrac{1}{2a} \cdot \ln \bigg |\cfrac{u + a}{u - a} \bigg | + \text{C}
• \displaystyle \int \cfrac{du}{\sqrt{a^{2} - u^{2}}} = \sin^{-1} \cfrac{u}{a} + \text{C}, \qquad a > 0
• \displaystyle \int \cfrac{du}{\sqrt{u^{2} - a^{2}}} = \ln \big ( u + \sqrt{u^{2} - a^{2}} \big ) + \text{C}
• \displaystyle \int \cfrac{du}{\sqrt{u^{2} + a^{2}}} = \ln \big ( u + \sqrt{u^{2} + a^{2}} \big ) + \text{C} = \sinh^{-1} \cfrac{u}{a} + \text{C}
• \displaystyle \int \cfrac{du}{u \cdot \sqrt{u^{2} - a^{2}}} = \cfrac{1}{a} \cdot \sec^{-1} \bigg | \cfrac{u}{a} \bigg | + \text{C}
• \displaystyle \int \cfrac{du}{u \cdot \sqrt{u^{2} + a^{2}}} = - \cfrac{1}{a} \cdot \ln \Bigg ( \cfrac{a + \sqrt{u^{2} + a^{2}}}{u} \Bigg ) + \text{C}
• \displaystyle \int \cfrac{du}{u \cdot \sqrt{a^{2} - u^{2}}} = - \cfrac{1}{a} \cdot \ln \Bigg ( \cfrac{a + \sqrt{a^{2} - u^{2}}}{u} \Bigg ) + \text{C}

## Integration formula by parts

• \displaystyle \int u \cdot dv = u \cdot v - \int v \cdot du

## Formulas of trigonometric integrals

#### Integral square sine

• \displaystyle \int \sin^{2}u \cdot du = \cfrac{1}{2} \cdot u - \cfrac{1}{4} \cdot \sin (2u) + \text{C} = \cfrac{1}{2} \cdot (u - \sin u \cdot \cos u) + \text{C}

#### Integral of square cosine

• \displaystyle \int \cos^{2}u \cdot du = \cfrac{1}{2} \cdot u + \cfrac{1}{4} \cdot \sin(2u) + \text{C} = \cfrac{1}{2} \cdot (u + \sin u \cdot \cos u)

#### Integral square tangent

• \displaystyle \int \tan^{2}u \cdot du = \tan u - u + \text{C}

#### Integral square cotangent

• \displaystyle \int \cot^{2}u \cdot du = - \cot u - u + \text{C}

#### Integral square secant

• \displaystyle \int \sec^{2}u \cdot du = \tan u + \text{C}

#### Integral square cosecant

• \displaystyle \int \csc^{2}u \cdot du = -\cot u + \text{C}

#### Integral cubic sinus

• \displaystyle \int \sin^{3}u \cdot du = - \cfrac{1}{3} \cdot (2 + \sin^{2} u) \cdot \cos u + \text{C}

#### Integral cubic cosine

• \displaystyle \int \cos^{3}u \cdot du = \cfrac{1}{3} \cdot (2 + \cos^{2} u) \cdot \sin u + \text{C}

#### Integral cubic tangent

• \displaystyle \int \tan^{3}u \cdot du = \cfrac{1}{2} \cdot \tan^{2}u + \ln |\cos u| + \text{C}

#### Integral of cubic cotangent

• \displaystyle \int \cot^{3}u \cdot du =  - \cfrac{1}{2}\cdot \cot^{2}u - \ln |\sin u| + \text{C}

#### Integral cubic secant

• \displaystyle \int \sec^{3}u \cdot du = \cfrac{1}{2} \cdot \sec u \cdot \tan u + \cfrac{1}{2} \cdot \ln |\sec u + \tan u| + \text{C}

#### Integral cubic cosecant

• \displaystyle \int \csc^{3}u \cdot du = - \cfrac{1}{2} \cdot \csc u \cdot \cot u + \cfrac{1}{2} \cdot \ln |\csc u - \cot u| + \text{C}

## Several integrals

• \displaystyle \int \sin au \cdot \sin bu \cdot du = \cfrac{\sin(a - b)\cdot u}{2 \cdot (a - b)} - \cfrac{\sin(a + b) \cdot u}{2 \cdot (a+b)}+ \text{C}
• \displaystyle \int \cos au \cdot \cos bu \cdot du = \cfrac{\sin(a - b) \cdot u}{2 \cdot (a - b)} + \cfrac{\sin(a + b) \cdot u}{2 \cdot (a+b)} + \text{C}
• \displaystyle \int \sin au \cdot \cos bu \cdot du = - \cfrac{\cos(a - b) \cdot u}{2 \cdot (a - b)} - \cfrac{\cos(a + b) \cdot u}{2 \cdot (a + b)} + \text{C}
• \displaystyle \int u \cdot \sin u \cdot du = \sin u - u \cdot \cos u + \text{C}
• \displaystyle \int u \cdot \cos u \cdot du = \cos u + u \cdot \sin u + \text{C}

## Formulas of trigonometric reduction integrals

• \displaystyle \int \sin^{n}u \cdot du = - \cfrac{1}{n} \cdot \sin^{n - 1} u \cdot \cos u + \displaystyle \cfrac{n-1}{n} \cdot \int \sin^{n-2}u \cdot du + \text{C}
• \displaystyle \int \cos^{n}u \cdot du = \cfrac{1}{n} \cdot \cos^{n-1} u \cdot \sin u + \displaystyle \cfrac{n-1}{n} \cdot \int \cos^{n-2}u \cdot du + \text{C}
• \displaystyle \int \tan^{n}u \cdot du = \displaystyle \cfrac{1}{1-n} \cdot \tan^{n - 1}u - \int \tan^{n - 2} u \cdot du + \text{C}
• \displaystyle \int \cot^{n}u \cdot du = \displaystyle - \cfrac{1}{n - 1} \cdot \cos^{n - 1}u - \displaystyle \int \cot^{n-2}u \cdot du + \text{C}
• \displaystyle \int \sec^{n}u \cdot du = \displaystyle \cfrac{1}{n-1} \cdot \tan u \cdot \sec^{n - 2}u + \displaystyle \cfrac{n-2}{n-1} \cdot \int \sec^{n-2}u \cdot du + \text{C}
• \displaystyle \int \csc^{n}u \cdot du = \displaystyle - \cfrac{1}{n-1} \cdot \cot u \cdot \csc^{n - 2}u + \displaystyle \cfrac{n-2}{n-1} \cdot \int \csc^{n - 2}u \cdot du + \text{C}
• \displaystyle \int u^{n} \cdot \sin u \cdot du = \displaystyle - u^{n} \cdot \cos u + n \cdot \int u^{n - 1} \cdot \cos u \cdot du + \text{C}
• \displaystyle \int u^{n} \cdot \cos u \cdot du = \displaystyle u^{n} \cdot \sin u - n \cdot \int u^{n - 1} \cdot \sin u \cdot du + \text{C}
• \displaystyle \int \sin^{n}u \cdot \cos^{m}u \cdot du = \displaystyle - \cfrac{\sin^{n - 1}u \cdot \cos^{m + 1} u}{n + m} + \cfrac{n - 1}{n + m} \cdot \int \sin^{n-2}u \cdot \cos^{m}u \cdot du + \text{C} = \displaystyle \cfrac{\sin^{n + 1}u \cdot \cos^{m - 1} u}{n + m} + \cfrac{m-1}{n+m} \cdot \int \sin^{n}u \cdot \cos^{m-2}u \cdot du + \text{C}

## Formulas of inverse trigonometric integrals

• \displaystyle \int \sin^{-1}u \cdot du = u \cdot \sin^{-1}u + \sqrt{1 - u^{2}} + \text{C}
• \displaystyle \int \cos^{-1}u \cdot du = u \cdot \cos^{-1}u - \sqrt{1 - u^{2}} + \text{C}
• \displaystyle \int \tan^{-1}u \cdot du = u \cdot \tan^{-1}u - \cfrac{1}{2} \cdot \ln \big ( 1+u^{2} \big ) + \text{C}
• \displaystyle \int u \cdot \sin^{-1}u \cdot du = \cfrac{2 \cdot u^{2} - 1}{4} \cdot \sin^{-1} u + \cfrac{u \cdot \sqrt{1 - u^{2}}}{4} + \text{C}
• \displaystyle \int u \cdot \cos^{-1}u \cdot du = \cfrac{2 \cdot u^{2} - 1}{4} \cdot \cos^{-1} u - \cfrac{ u \cdot \sqrt{1-u^{2}}}{4} + \text{C}
• \displaystyle \int u \cdot \tan^{-1}u \cdot du = \cfrac{u^{2} + 1}{4} \cdot \tan^{-1}u - \cfrac{u}{2} + \text{C}
• \displaystyle \int u^{n} \cdot \sin^{-1} u \cdot du = \displaystyle \cfrac{1}{n + 1} \cdot \Bigg [ u^{n+1} \cdot \sin^{-1}u - \int \cfrac{u^{n + 1} \cdot du}{\sqrt{1 - u^{2}}} \Bigg ] + \text{C}, \quad n \neq -1
• \displaystyle \int \ u^{n} \cdot \cos^{-1}u \cdot du = \displaystyle \cfrac{1}{n + 1} \cdot \Bigg [ u^{n + 1} \cdot \cos^{-1}u + \int \cfrac{u^{n + 1} \cdot du}{ \sqrt{1 - u^{2}}} \Bigg ] + \text{C}, \quad n \neq -1
• \displaystyle \int u^{n} \cdot \tan^{-1}u \cdot du = \displaystyle \cfrac{1}{n + 1} \cdot \Bigg [ u^{n + 1} \cdot \tan^{-1}u - \int \cfrac{u^{n + 1} \cdot du}{1 + u^{2}} \Bigg ] + \text{C}, \quad n \neq -1

## Substantial substitutions of the integrals

• It has u = a \cdot x + b
\displaystyle \int F(a \cdot x + b) \cdot dx = \displaystyle \cfrac{1}{a} \cdot \int F(u) \cdot du + \text{C}
• It has u = \sqrt{a \cdot x + b}
\displaystyle \int F \big ( \sqrt{a \cdot x + b} \big ) \cdot dx = \displaystyle \cfrac{2}{a} \cdot \int u \cdot F(u) \cdot du + \text{C}
• It has u = (a \cdot x + b)^{1/n}
\displaystyle \int F \big ( (a \cdot x + b)^{1/n} \big ) \cdot dx = \displaystyle \cfrac{n}{a} \cdot \int u^{n-1} \cdot F(u) \cdot du + \text{C}
• It has u = a \cdot \sin u
\displaystyle \int F \big ( \sqrt{a^{2} - x^{2}} \big ) \cdot dx = \displaystyle a \cdot \int F(a \cdot \cos u) \cdot \cos u \cdot du + \text{C}
• It has u = a \cdot \tan u
\displaystyle \int F \big ( \sqrt{x^{2} + a^{2}} \big ) \cdot dx = \displaystyle a \cdot \int F(a \cdot \sec u) \cdot \sec^{2} u \cdot du + \text{C}
• It has u = a \cdot \sec u
\displaystyle \int \ F \big ( \sqrt{x^{2} - a^{2}} \big ) \cdot dx = \displaystyle a \cdot \int \ F(a \cdot \tan u) \cdot \sec u \cdot \tan u \cdot du + \text{C}
• It has u = e^{a \cdot x}
\displaystyle \int F(e^{a\cdot x}) \cdot dx = \displaystyle \cfrac{1}{a} \cdot \int \cfrac{F(u)}{u} \cdot du + \text{C}
• It has u = \ln x
\displaystyle \int F(\ln x) \cdot dx = \displaystyle \cfrac{1}{a} \cdot \int F(u) \cdot e^{u} \cdot du + \text{C}
• It has u = \sin^{-1} \cfrac{x}{a}
\displaystyle \int F \bigg ( \sin^{-1} \cfrac{x}{a} \bigg ) \cdot dx = \displaystyle a \cdot \int F(u) \cdot \cos u \cdot du + \text{C}