What is the integral of ((5x+3)/((x+2)^2(x-1)^2)) ?

The integral of ((5x+3)/((x+2)^2(x-1)^2)) is 1/27 ln|x+2|+7/(9(x+2))-1/27 ln|x-1|-8/(9(x-1))+C

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integral of ((5x+3)/((x+2)^2(x-1)^2))

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

\frac{1}{27} ln ∣ x + 2 ∣ + \frac{7}{9 ( x + 2 )} - \frac{1}{27} ln ∣ x - 1 ∣ - \frac{8}{9 ( x - 1 )} + C

Solution steps

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

Take the partial fraction of

\frac{5 x + 3}{( x + 2 ) ^{2} ( x - 1 ) ^{2}} : \frac{1}{27 ( x + 2 )} - \frac{7}{9 ( x + 2 ) ^{2}} - \frac{1}{27 ( x - 1 )} + \frac{8}{9 ( x - 1 ) ^{2}}

= \int \frac{1}{27 ( x + 2 )} - \frac{7}{9 ( x + 2 ) ^{2}} - \frac{1}{27 ( x - 1 )} + \frac{8}{9 ( x - 1 ) ^{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{1}{27 ( x + 2 )} d x - \int \frac{7}{9 ( x + 2 ) ^{2}} d x - \int \frac{1}{27 ( x - 1 )} d x + \int \frac{8}{9 ( x - 1 ) ^{2}} d x

\int \frac{1}{27 ( x + 2 )} d x = \frac{1}{27} ln ∣ x + 2 ∣

\int \frac{7}{9 ( x + 2 ) ^{2}} d x = - \frac{7}{9 ( x + 2 )}

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

\int \frac{8}{9 ( x - 1 ) ^{2}} d x = - \frac{8}{9 ( x - 1 )}

= \frac{1}{27} ln ∣ x + 2 ∣ - (- \frac{7}{9 ( x + 2 )}) - \frac{1}{27} ln ∣ x - 1 ∣ - \frac{8}{9 ( x - 1 )}

Simplify

= \frac{1}{27} ln ∣ x + 2 ∣ + \frac{7}{9 ( x + 2 )} - \frac{1}{27} ln ∣ x - 1 ∣ - \frac{8}{9 ( x - 1 )}

Add a constant to the solution

= \frac{1}{27} ln ∣ x + 2 ∣ + \frac{7}{9 ( x + 2 )} - \frac{1}{27} ln ∣ x - 1 ∣ - \frac{8}{9 ( x - 1 )} + C

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What is the integral of ((5x+3)/((x+2)^2(x-1)^2)) ?
The integral of ((5x+3)/((x+2)^2(x-1)^2)) is 1/27 ln|x+2|+7/(9(x+2))-1/27 ln|x-1|-8/(9(x-1))+C