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

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

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

\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}} + C

Solution steps

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

Apply Integration By Parts

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

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

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

Simplify

= \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}}

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}} + C

5 y

\frac{d}{d x} (\frac{d y}{d x}) + y = 7 sin (x)

\int \frac{0.8 ( 3 + ln ( u ) ) ^{3}}{u} d u

y = 2 x^{3} - 8 x, (2, 0)

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