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

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

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

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

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

Solution steps

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

Apply Integral Substitution

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

Take the constant out:

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

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

Factor

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

= x \cdot \int \frac{- ( u + 1 ) ( u - 1 )}{x ^{2} ( u ^{2} + 1 ) ^{2}} d u

Take the constant out:

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

= x \frac{- 1}{x ^{2}} \cdot \int \frac{( u + 1 ) ( u - 1 )}{( u ^{2} + 1 ) ^{2}} d u

Simplify

= x (- \frac{1}{x ^{2}}) \cdot \int \frac{( u + 1 ) ( u - 1 )}{( u ^{2} + 1 ) ^{2}} d u

Take the partial fraction of

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

= x (- \frac{1}{x ^{2}}) \cdot \int \frac{1}{u ^{2} + 1} - \frac{2}{( u ^{2} + 1 ) ^{2}} d u

Apply the Sum Rule:

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

= x (- \frac{1}{x ^{2}}) (\int \frac{1}{u ^{2} + 1} d u - \int \frac{2}{( u ^{2} + 1 ) ^{2}} d u)

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

\int \frac{2}{( u ^{2} + 1 ) ^{2}} d u = arctan (u) + \frac{u}{1 + u ^{2}}

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

Substitute back

u = \frac{y}{x}

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

Simplify

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

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

Add a constant to the solution

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

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