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

The integral of (x^2-1)/((x-2)^2(x+3)) is 17/25 ln|x-2|-3/(5(x-2))+8/25 ln|x+3|+C

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

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

\frac{17}{25} ln ∣ x - 2 ∣ - \frac{3}{5 ( x - 2 )} + \frac{8}{25} ln ∣ x + 3 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x ^{2} - 1}{( x - 2 ) ^{2} ( x + 3 )} : \frac{17}{25 ( x - 2 )} + \frac{3}{5 ( x - 2 ) ^{2}} + \frac{8}{25 ( x + 3 )}

= \int \frac{17}{25 ( x - 2 )} + \frac{3}{5 ( x - 2 ) ^{2}} + \frac{8}{25 ( x + 3 )} 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{17}{25 ( x - 2 )} d x + \int \frac{3}{5 ( x - 2 ) ^{2}} d x + \int \frac{8}{25 ( x + 3 )} d x

\int \frac{17}{25 ( x - 2 )} d x = \frac{17}{25} ln ∣ x - 2 ∣

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

\int \frac{8}{25 ( x + 3 )} d x = \frac{8}{25} ln ∣ x + 3 ∣

= \frac{17}{25} ln ∣ x - 2 ∣ - \frac{3}{5 ( x - 2 )} + \frac{8}{25} ln ∣ x + 3 ∣

Add a constant to the solution

= \frac{17}{25} ln ∣ x - 2 ∣ - \frac{3}{5 ( x - 2 )} + \frac{8}{25} ln ∣ x + 3 ∣ + C

\frac{\partial}{\partial x} (\frac{x}{( x + y ) ^{2}})

\int x e^{x^{5}} d x

(8 - 1) (82)

\frac{d}{d x} (36 x)

What is the integral of (x^2-1)/((x-2)^2(x+3)) ?
The integral of (x^2-1)/((x-2)^2(x+3)) is 17/25 ln|x-2|-3/(5(x-2))+8/25 ln|x+3|+C