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

The integral of (-y)/((x^2+y^2)^2) is 1/(2(x^2+y^2))+C

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= - \int \frac{1}{2 u ^{2}} d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= - \frac{1}{2} (- \frac{1}{u})

Substitute back

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

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

Simplify

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

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

Add a constant to the solution

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

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