What is the integral of (x^2)/((x+2)^3) ?

The integral of (x^2)/((x+2)^3) is ln|x+2|+4/(x+2)-2/((x+2)^2)+C

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integral of (x^2)/((x+2)^3)

integral

\frac{x ^{2}}{( x + 2 ) ^{3}}

ln ∣ x + 2 ∣ + \frac{4}{x + 2} - \frac{2}{( x + 2 ) ^{2}} + C

Solution steps

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

Apply u-substitution

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

Expand

\frac{( u - 2 ) ^{2}}{u ^{3}} : \frac{1}{u} - \frac{4}{u ^{2}} + \frac{4}{u ^{3}}

= \int \frac{1}{u} - \frac{4}{u ^{2}} + \frac{4}{u ^{3}} 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} d u - \int \frac{4}{u ^{2}} d u + \int \frac{4}{u ^{3}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{4}{u ^{2}} d u = - \frac{4}{u}

\int \frac{4}{u ^{3}} d u = - \frac{2}{u ^{2}}

= ln ∣ u ∣ - (- \frac{4}{u}) - \frac{2}{u ^{2}}

Substitute back

u = x + 2

= ln ∣ x + 2 ∣ - (- \frac{4}{x + 2}) - \frac{2}{( x + 2 ) ^{2}}

Simplify

= ln ∣ x + 2 ∣ + \frac{4}{x + 2} - \frac{2}{( x + 2 ) ^{2}}

Add a constant to the solution

= ln ∣ x + 2 ∣ + \frac{4}{x + 2} - \frac{2}{( x + 2 ) ^{2}} + C

What is the integral of (x^2)/((x+2)^3) ?
The integral of (x^2)/((x+2)^3) is ln|x+2|+4/(x+2)-2/((x+2)^2)+C