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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{2 u} d u

= x \cdot \int \frac{1}{2 u} d u

Take the constant out:

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

= x \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= x \frac{1}{2} ln ∣ u ∣

Substitute back

u = x^{2} + y^{2}

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

Add a constant to the solution

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

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