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

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

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integral of arctan(y/x)

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

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

Solution steps

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

Apply Integration By Parts

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

= 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}

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

Simplify

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

Add a constant to the solution

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

\int \frac{ln ( x ^{2} )}{x ^{4}} d x

\int \frac{( x ^{3} + 1 )}{x ^{4} + 4 x} d x

2

\int 3^{u} d u

\int \frac{2 sin ( x ) - 5 cos ( x )}{cos ( x )} d x

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