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

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

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integral of (x+y)/(x^2+y^2)

\int \frac{x + y}{x ^{2} + y ^{2}} d y

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

Solution steps

\int \frac{x + y}{x ^{2} + y ^{2}} d y

Expand

\frac{x + y}{x ^{2} + y ^{2}} : \frac{x}{x ^{2} + y ^{2}} + \frac{y}{x ^{2} + y ^{2}}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Add a constant to the solution

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

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