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

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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

= 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

\int e^{\frac{5 x}{2}} d x

\int x d x d x

\int \frac{17 x + 4}{3 x ^{2} - 7 x - 6} d x

\int 2 x - \frac{1}{5} d x

\int x e^{2 y} d x

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