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

The integral of (3x+2)/((x+2)^3(x-1)) is -5/27 ln|x+2|+5/(9(x+2))-2/(3(x+2)^2)+5/27 ln|x-1|+C

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

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

- \frac{5}{27} ln ∣ x + 2 ∣ + \frac{5}{9 ( x + 2 )} - \frac{2}{3 ( x + 2 ) ^{2}} + \frac{5}{27} ln ∣ x - 1 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{3 x + 2}{( x + 2 ) ^{3} ( x - 1 )} : - \frac{5}{27 ( x + 2 )} - \frac{5}{9 ( x + 2 ) ^{2}} + \frac{4}{3 ( x + 2 ) ^{3}} + \frac{5}{27 ( x - 1 )}

= \int - \frac{5}{27 ( x + 2 )} - \frac{5}{9 ( x + 2 ) ^{2}} + \frac{4}{3 ( x + 2 ) ^{3}} + \frac{5}{27 ( x - 1 )} 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{5}{27 ( x + 2 )} d x - \int \frac{5}{9 ( x + 2 ) ^{2}} d x + \int \frac{4}{3 ( x + 2 ) ^{3}} d x + \int \frac{5}{27 ( x - 1 )} d x

\int \frac{5}{27 ( x + 2 )} d x = \frac{5}{27} ln ∣ x + 2 ∣

\int \frac{5}{9 ( x + 2 ) ^{2}} d x = - \frac{5}{9 ( x + 2 )}

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

\int \frac{5}{27 ( x - 1 )} d x = \frac{5}{27} ln ∣ x - 1 ∣

= - \frac{5}{27} ln ∣ x + 2 ∣ - (- \frac{5}{9 ( x + 2 )}) - \frac{2}{3 ( x + 2 ) ^{2}} + \frac{5}{27} ln ∣ x - 1 ∣

Simplify

= - \frac{5}{27} ln ∣ x + 2 ∣ + \frac{5}{9 ( x + 2 )} - \frac{2}{3 ( x + 2 ) ^{2}} + \frac{5}{27} ln ∣ x - 1 ∣

Add a constant to the solution

= - \frac{5}{27} ln ∣ x + 2 ∣ + \frac{5}{9 ( x + 2 )} - \frac{2}{3 ( x + 2 ) ^{2}} + \frac{5}{27} ln ∣ x - 1 ∣ + C

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What is the integral of (3x+2)/((x+2)^3(x-1)) ?
The integral of (3x+2)/((x+2)^3(x-1)) is -5/27 ln|x+2|+5/(9(x+2))-2/(3(x+2)^2)+5/27 ln|x-1|+C