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

The integral of (x^2+2)/((x+1)^3(x-2)) is -2/9 ln|x+1|-1/(3(x+1))+1/(2(x+1)^2)+2/9 ln|x-2|+C

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

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

- \frac{2}{9} ln ∣ x + 1 ∣ - \frac{1}{3 ( x + 1 )} + \frac{1}{2 ( x + 1 ) ^{2}} + \frac{2}{9} ln ∣ x - 2 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x ^{2} + 2}{( x + 1 ) ^{3} ( x - 2 )} : - \frac{2}{9 ( x + 1 )} + \frac{1}{3 ( x + 1 ) ^{2}} - \frac{1}{( x + 1 ) ^{3}} + \frac{2}{9 ( x - 2 )}

= \int - \frac{2}{9 ( x + 1 )} + \frac{1}{3 ( x + 1 ) ^{2}} - \frac{1}{( x + 1 ) ^{3}} + \frac{2}{9 ( x - 2 )} 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{2}{9 ( x + 1 )} d x + \int \frac{1}{3 ( x + 1 ) ^{2}} d x - \int \frac{1}{( x + 1 ) ^{3}} d x + \int \frac{2}{9 ( x - 2 )} d x

\int \frac{2}{9 ( x + 1 )} d x = \frac{2}{9} ln ∣ x + 1 ∣

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

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

\int \frac{2}{9 ( x - 2 )} d x = \frac{2}{9} ln ∣ x - 2 ∣

= - \frac{2}{9} ln ∣ x + 1 ∣ - \frac{1}{3 ( x + 1 )} - (- \frac{1}{2 ( x + 1 ) ^{2}}) + \frac{2}{9} ln ∣ x - 2 ∣

Simplify

= - \frac{2}{9} ln ∣ x + 1 ∣ - \frac{1}{3 ( x + 1 )} + \frac{1}{2 ( x + 1 ) ^{2}} + \frac{2}{9} ln ∣ x - 2 ∣

Add a constant to the solution

= - \frac{2}{9} ln ∣ x + 1 ∣ - \frac{1}{3 ( x + 1 )} + \frac{1}{2 ( x + 1 ) ^{2}} + \frac{2}{9} ln ∣ x - 2 ∣ + C

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What is the integral of (x^2+2)/((x+1)^3(x-2)) ?
The integral of (x^2+2)/((x+1)^3(x-2)) is -2/9 ln|x+1|-1/(3(x+1))+1/(2(x+1)^2)+2/9 ln|x-2|+C