What is the integral of (x^2)/(x^2+5x-24) ?

The integral of (x^2)/(x^2+5x-24) is x+9/11 ln|x-3|-64/11 ln|x+8|+C

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integral of (x^2)/(x^2+5x-24)

\int \frac{x ^{2}}{x ^{2} + 5 x - 24} d x

x + \frac{9}{11} ln ∣ x - 3 ∣ - \frac{64}{11} ln ∣ x + 8 ∣ + C

Solution steps

\int \frac{x ^{2}}{x ^{2} + 5 x - 24} d x

Factor

\frac{x ^{2}}{x ^{2} + 5 x - 24}

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

Take the partial fraction of

\frac{x ^{2}}{( x - 3 ) ( x + 8 )} : 1 + \frac{9}{11 ( x - 3 )} - \frac{64}{11 ( x + 8 )}

= \int 1 + \frac{9}{11 ( x - 3 )} - \frac{64}{11 ( x + 8 )} 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 1 d x + \int \frac{9}{11 ( x - 3 )} d x - \int \frac{64}{11 ( x + 8 )} d x

\int 1 d x = x

\int \frac{9}{11 ( x - 3 )} d x = \frac{9}{11} ln ∣ x - 3 ∣

\int \frac{64}{11 ( x + 8 )} d x = \frac{64}{11} ln ∣ x + 8 ∣

= x + \frac{9}{11} ln ∣ x - 3 ∣ - \frac{64}{11} ln ∣ x + 8 ∣

Add a constant to the solution

= x + \frac{9}{11} ln ∣ x - 3 ∣ - \frac{64}{11} ln ∣ x + 8 ∣ + C

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What is the integral of (x^2)/(x^2+5x-24) ?
The integral of (x^2)/(x^2+5x-24) is x+9/11 ln|x-3|-64/11 ln|x+8|+C