What is the integral of (x^2)/((1+x)^3) ?

The integral of (x^2)/((1+x)^3) is

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integral of (x^2)/((1+x)^3)

\int \frac{x ^{2}}{( 1 + x ) ^{3}} d x

\frac{x ^{3}}{( 1 + x ) ^{3}} + ln ∣ x + 1 ∣ + \frac{3}{x + 1} - \frac{3}{2 ( x + 1 ) ^{2}} + \frac{1}{3 ( x + 1 ) ^{3}} + \frac{2 x ^{2}}{3 ( 1 + x ) ^{3}} - \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}} + C

Solution steps

\int \frac{x ^{2}}{( 1 + x ) ^{3}} d x

Apply Integration By Parts

: \frac{x ^{3}}{( 1 + x ) ^{3}} - \int \frac{x ^{2} ( - x + 2 )}{( 1 + x ) ^{4}} d x

= \frac{x ^{3}}{( 1 + x ) ^{3}} - \int \frac{x ^{2} ( - x + 2 )}{( 1 + x ) ^{4}} d x

\int \frac{x ^{2} ( - x + 2 )}{( 1 + x ) ^{4}} d x = - ln ∣ x + 1 ∣ - \frac{3}{x + 1} + \frac{3}{2 ( x + 1 ) ^{2}} - \frac{1}{3 ( x + 1 ) ^{3}} - \frac{2 x ^{2}}{3 ( 1 + x ) ^{3}} + \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}}

= \frac{x ^{3}}{( 1 + x ) ^{3}} - (- ln ∣ x + 1 ∣ - \frac{3}{x + 1} + \frac{3}{2 ( x + 1 ) ^{2}} - \frac{1}{3 ( x + 1 ) ^{3}} - \frac{2 x ^{2}}{3 ( 1 + x ) ^{3}} + \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}})

Simplify

= \frac{x ^{3}}{( 1 + x ) ^{3}} + ln ∣ x + 1 ∣ + \frac{3}{x + 1} - \frac{3}{2 ( x + 1 ) ^{2}} + \frac{1}{3 ( x + 1 ) ^{3}} + \frac{2 x ^{2}}{3 ( 1 + x ) ^{3}} - \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}}

Add a constant to the solution

= \frac{x ^{3}}{( 1 + x ) ^{3}} + ln ∣ x + 1 ∣ + \frac{3}{x + 1} - \frac{3}{2 ( x + 1 ) ^{2}} + \frac{1}{3 ( x + 1 ) ^{3}} + \frac{2 x ^{2}}{3 ( 1 + x ) ^{3}} - \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}} + C

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