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

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

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

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

- \frac{1}{x} + \frac{1}{x + 1} + C

Solution steps

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

Take the partial fraction of

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

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

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

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

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

Simplify

= - \frac{1}{x} + \frac{1}{x + 1}

Add a constant to the solution

= - \frac{1}{x} + \frac{1}{x + 1} + C

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