What is the integral of (3x^3+5x)/(x^2-x-2) ?

The integral of (3x^3+5x)/(x^2-x-2) is (3x^2)/2+3x+8/3 ln|x+1|+34/3 ln|x-2|+C

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integral of (3x^3+5x)/(x^2-x-2)

\int \frac{3 x ^{3} + 5 x}{x ^{2} - x - 2} d x

\frac{3 x ^{2}}{2} + 3 x + \frac{8}{3} ln ∣ x + 1 ∣ + \frac{34}{3} ln ∣ x - 2 ∣ + C

Solution steps

\int \frac{3 x ^{3} + 5 x}{x ^{2} - x - 2} d x

Long division

\frac{3 x ^{3} + 5 x}{x ^{2} - x - 2} : 3 x + 3 + \frac{14 x + 6}{x ^{2} - x - 2}

= \int 3 x + 3 + \frac{14 x + 6}{x ^{2} - x - 2} 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 3 x d x + \int 3 d x + \int \frac{14 x + 6}{x ^{2} - x - 2} d x

\int 3 x d x = \frac{3 x ^{2}}{2}

\int 3 d x = 3 x

\int \frac{14 x + 6}{x ^{2} - x - 2} d x = \frac{8}{3} ln ∣ x + 1 ∣ + \frac{34}{3} ln ∣ x - 2 ∣

= \frac{3 x ^{2}}{2} + 3 x + \frac{8}{3} ln ∣ x + 1 ∣ + \frac{34}{3} ln ∣ x - 2 ∣

Add a constant to the solution

= \frac{3 x ^{2}}{2} + 3 x + \frac{8}{3} ln ∣ x + 1 ∣ + \frac{34}{3} ln ∣ x - 2 ∣ + C

x \to - \infty lim (arccos (x))

\int \frac{x ^{2} - 15 L}{x + 7} d x

\int \frac{x ^{2}}{( 1 - 25 x ^{2} ) ^{\frac{3}{2}}} d x

\frac{1 - 3 x - x ^{2}}{x ^{2} - 3}

2 t^{3}

What is the integral of (3x^3+5x)/(x^2-x-2) ?
The integral of (3x^3+5x)/(x^2-x-2) is (3x^2)/2+3x+8/3 ln|x+1|+34/3 ln|x-2|+C