What is the integral of 1/((x+5)^2(x-4)) ?

The integral of 1/((x+5)^2(x-4)) is -1/81 ln|x+5|+1/(9(x+5))+1/81 ln|x-4|+C

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integral of 1/((x+5)^2(x-4))

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

- \frac{1}{81} ln ∣ x + 5 ∣ + \frac{1}{9 ( x + 5 )} + \frac{1}{81} ln ∣ x - 4 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{1}{( x + 5 ) ^{2} ( x - 4 )} : - \frac{1}{81 ( x + 5 )} - \frac{1}{9 ( x + 5 ) ^{2}} + \frac{1}{81 ( x - 4 )}

= \int - \frac{1}{81 ( x + 5 )} - \frac{1}{9 ( x + 5 ) ^{2}} + \frac{1}{81 ( x - 4 )} 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}{81 ( x + 5 )} d x - \int \frac{1}{9 ( x + 5 ) ^{2}} d x + \int \frac{1}{81 ( x - 4 )} d x

\int \frac{1}{81 ( x + 5 )} d x = \frac{1}{81} ln ∣ x + 5 ∣

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

\int \frac{1}{81 ( x - 4 )} d x = \frac{1}{81} ln ∣ x - 4 ∣

= - \frac{1}{81} ln ∣ x + 5 ∣ - (- \frac{1}{9 ( x + 5 )}) + \frac{1}{81} ln ∣ x - 4 ∣

Simplify

= - \frac{1}{81} ln ∣ x + 5 ∣ + \frac{1}{9 ( x + 5 )} + \frac{1}{81} ln ∣ x - 4 ∣

Add a constant to the solution

= - \frac{1}{81} ln ∣ x + 5 ∣ + \frac{1}{9 ( x + 5 )} + \frac{1}{81} ln ∣ x - 4 ∣ + C

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What is the integral of 1/((x+5)^2(x-4)) ?
The integral of 1/((x+5)^2(x-4)) is -1/81 ln|x+5|+1/(9(x+5))+1/81 ln|x-4|+C