What is the integral of arctan(x/y) ?

The integral of arctan(x/y) is xarctan(x/y)-y 1/2 ln|x^2+y^2|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of arctan(x/y)

\int arctan (\frac{x}{y}) d x

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

Solution steps

\int arctan (\frac{x}{y}) d x

Apply Integration By Parts

= x arctan (\frac{x}{y}) - \int \frac{y x}{x ^{2} + y ^{2}} d x

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

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

Add a constant to the solution

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

What is the integral of arctan(x/y) ?
The integral of arctan(x/y) is xarctan(x/y)-y 1/2 ln|x^2+y^2|+C