What is the integral of (5x-7x^2+4)/((x-2)^2(x+1)) ?

The integral of (5x-7x^2+4)/((x-2)^2(x+1)) is -55/9 ln|x-2|+(14)/(3(x-2))-8/9 ln|x+1|+C

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integral of (5x-7x^2+4)/((x-2)^2(x+1))

\int \frac{5 x - 7 x ^{2} + 4}{( x - 2 ) ^{2} ( x + 1 )} d x

- \frac{55}{9} ln ∣ x - 2 ∣ + \frac{14}{3 ( x - 2 )} - \frac{8}{9} ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{5 x - 7 x ^{2} + 4}{( x - 2 ) ^{2} ( x + 1 )} d x

Take the partial fraction of

\frac{5 x - 7 x ^{2} + 4}{( x - 2 ) ^{2} ( x + 1 )} : - \frac{55}{9 ( x - 2 )} - \frac{14}{3 ( x - 2 ) ^{2}} - \frac{8}{9 ( x + 1 )}

= \int - \frac{55}{9 ( x - 2 )} - \frac{14}{3 ( x - 2 ) ^{2}} - \frac{8}{9 ( 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{55}{9 ( x - 2 )} d x - \int \frac{14}{3 ( x - 2 ) ^{2}} d x - \int \frac{8}{9 ( x + 1 )} d x

\int \frac{55}{9 ( x - 2 )} d x = \frac{55}{9} ln ∣ x - 2 ∣

\int \frac{14}{3 ( x - 2 ) ^{2}} d x = - \frac{14}{3 ( x - 2 )}

\int \frac{8}{9 ( x + 1 )} d x = \frac{8}{9} ln ∣ x + 1 ∣

= - \frac{55}{9} ln ∣ x - 2 ∣ - (- \frac{14}{3 ( x - 2 )}) - \frac{8}{9} ln ∣ x + 1 ∣

Simplify

= - \frac{55}{9} ln ∣ x - 2 ∣ + \frac{14}{3 ( x - 2 )} - \frac{8}{9} ln ∣ x + 1 ∣

Add a constant to the solution

= - \frac{55}{9} ln ∣ x - 2 ∣ + \frac{14}{3 ( x - 2 )} - \frac{8}{9} ln ∣ x + 1 ∣ + C

What is the integral of (5x-7x^2+4)/((x-2)^2(x+1)) ?
The integral of (5x-7x^2+4)/((x-2)^2(x+1)) is -55/9 ln|x-2|+(14)/(3(x-2))-8/9 ln|x+1|+C