What is the integral of (2x+1)/((2x^2+2x)^2) ?

The integral of (2x+1)/((2x^2+2x)^2) is -1/(4x)+1/(4(x+1))+C

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integral of (2x+1)/((2x^2+2x)^2)

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

- \frac{1}{4 x} + \frac{1}{4 ( x + 1 )} + C

Solution steps

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

Take the partial fraction of

\frac{2 x + 1}{( 2 x ^{2} + 2 x ) ^{2}} : \frac{1}{4 x ^{2}} - \frac{1}{4 ( x + 1 ) ^{2}}

= \int \frac{1}{4 x ^{2}} - \frac{1}{4 ( x + 1 ) ^{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{1}{4 x ^{2}} d x - \int \frac{1}{4 ( x + 1 ) ^{2}} d x

\int \frac{1}{4 x ^{2}} d x = - \frac{1}{4 x}

\int \frac{1}{4 ( x + 1 ) ^{2}} d x = - \frac{1}{4 ( x + 1 )}

= - \frac{1}{4 x} - (- \frac{1}{4 ( x + 1 )})

Simplify

= - \frac{1}{4 x} + \frac{1}{4 ( x + 1 )}

Add a constant to the solution

= - \frac{1}{4 x} + \frac{1}{4 ( x + 1 )} + C

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What is the integral of (2x+1)/((2x^2+2x)^2) ?
The integral of (2x+1)/((2x^2+2x)^2) is -1/(4x)+1/(4(x+1))+C