What is the integral of (2x+8)/(x^3+x^2) ?

The integral of (2x+8)/(x^3+x^2) is -6ln|x|-8/x+6ln|x+1|+C

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integral of (2x+8)/(x^3+x^2)

\int \frac{2 x + 8}{x ^{3} + x ^{2}} d x

- 6 ln ∣ x ∣ - \frac{8}{x} + 6 ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{2 x + 8}{x ^{3} + x ^{2}} d x

Take the partial fraction of

\frac{2 x + 8}{x ^{3} + x ^{2}} : - \frac{6}{x} + \frac{8}{x ^{2}} + \frac{6}{x + 1}

= \int - \frac{6}{x} + \frac{8}{x ^{2}} + \frac{6}{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{6}{x} d x + \int \frac{8}{x ^{2}} d x + \int \frac{6}{x + 1} d x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

\int \frac{8}{x ^{2}} d x = - \frac{8}{x}

\int \frac{6}{x + 1} d x = 6 ln ∣ x + 1 ∣

= - 6 ln ∣ x ∣ - \frac{8}{x} + 6 ln ∣ x + 1 ∣

Add a constant to the solution

= - 6 ln ∣ x ∣ - \frac{8}{x} + 6 ln ∣ x + 1 ∣ + C

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What is the integral of (2x+8)/(x^3+x^2) ?
The integral of (2x+8)/(x^3+x^2) is -6ln|x|-8/x+6ln|x+1|+C