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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

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