What is the integral of (6x+6)/((x^2+2x)^2) ?

The integral of (6x+6)/((x^2+2x)^2) is -3/(2x)+3/(2(x+2))+C

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integral of (6x+6)/((x^2+2x)^2)

\int \frac{6 x + 6}{( x ^{2} + 2 x ) ^{2}} d x

- \frac{3}{2 x} + \frac{3}{2 ( x + 2 )} + C

Solution steps

\int \frac{6 x + 6}{( x ^{2} + 2 x ) ^{2}} d x

Take the partial fraction of

\frac{6 x + 6}{( x ^{2} + 2 x ) ^{2}} : \frac{3}{2 x ^{2}} - \frac{3}{2 ( x + 2 ) ^{2}}

= \int \frac{3}{2 x ^{2}} - \frac{3}{2 ( x + 2 ) ^{2}} d x

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{3}{2 x ^{2}} d x - \int \frac{3}{2 ( x + 2 ) ^{2}} d x

\int \frac{3}{2 x ^{2}} d x = - \frac{3}{2 x}

\int \frac{3}{2 ( x + 2 ) ^{2}} d x = - \frac{3}{2 ( x + 2 )}

= - \frac{3}{2 x} - (- \frac{3}{2 ( x + 2 )})

Simplify

= - \frac{3}{2 x} + \frac{3}{2 ( x + 2 )}

Add a constant to the solution

= - \frac{3}{2 x} + \frac{3}{2 ( x + 2 )} + C

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What is the integral of (6x+6)/((x^2+2x)^2) ?
The integral of (6x+6)/((x^2+2x)^2) is -3/(2x)+3/(2(x+2))+C