What is the integral of aarctan(bx) ?

The integral of aarctan(bx) is a(xarctan(bx)-1/(2b)ln|b^2x^2+1|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of aarctan(bx)

\int a arctan (b x) d x

a (x arctan (b x) - \frac{1}{2 b} ln b^{2} x^{2} + 1) + C

Solution steps

\int a arctan (b x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= a \cdot \int arctan (b x) d x

Apply Integration By Parts

= a (x arctan (b x) - \int \frac{b x}{b ^{2} x ^{2} + 1} d x)

\int \frac{b x}{b ^{2} x ^{2} + 1} d x = \frac{1}{2 b} ln b^{2} x^{2} + 1

= a (x arctan (b x) - \frac{1}{2 b} ln b^{2} x^{2} + 1)

Add a constant to the solution

= a (x arctan (b x) - \frac{1}{2 b} ln b^{2} x^{2} + 1) + C

What is the integral of aarctan(bx) ?
The integral of aarctan(bx) is a(xarctan(bx)-1/(2b)ln|b^2x^2+1|)+C