What is the integral of 2arctan(y/x) ?

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

integral of 2arctan(y/x)

\int 2 arctan (\frac{y}{x}) d x

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

Solution steps

\int 2 arctan (\frac{y}{x}) d x

Take the constant out:

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

= 2 \cdot \int arctan (\frac{y}{x}) d x

Apply Integration By Parts

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

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

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{x ^{3} + 3 x ^{2} + 5 x + 4}{x ^{2} + 3 x + 2} d x

in v erse l a pl a ce \frac{( 10 )}{5 \cdot s ^{2} + 7 s}

\int \frac{4 y ^{2} - 7 y - 12}{y ( y + 2 ) ( y - 3 )} d y

d er i v a t i v e c y^{- 7}

\int 3 e^{\frac{x}{3}} d x

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