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

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

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

integral

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

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

Solution steps

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

Apply u-substitution

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

Take the partial fraction of

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

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

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

\int \frac{1}{u - 1} d u = ln ∣ u - 1 ∣

\int \frac{1}{u} d u = ln ∣ u ∣

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

= ln ∣ u - 1 ∣ - ln ∣ u ∣ - (- \frac{1}{u})

Substitute back

u = x + 1

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

Simplify

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

Add a constant to the solution

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

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