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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Expand

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

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

\int 1 d u = u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

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

= u - 4 ln ∣ u ∣ - \frac{4}{u}

Substitute back

u = x + 2

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

Add a constant to the solution

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

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