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

The integral of (x^3+2x^2+x+2)/((x+1)^2) is -(x^3+2x^2+x+2}{x+1}+\frac{3x^2)/2+x+C

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (x^3+2x^2+x+2)/((x+1)^2) ?
The integral of (x^3+2x^2+x+2)/((x+1)^2) is -(x^3+2x^2+x+2}{x+1}+\frac{3x^2)/2+x+C