What is the integral of (x^3+3x)/(x^2-3x-10) ?

The integral of (x^3+3x)/(x^2-3x-10) is (x^2)/2+3x+2ln|x+2|+20ln|x-5|+C

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integral of (x^3+3x)/(x^2-3x-10)

\int \frac{x ^{3} + 3 x}{x ^{2} - 3 x - 10} d x

\frac{x ^{2}}{2} + 3 x + 2 ln ∣ x + 2 ∣ + 20 ln ∣ x - 5 ∣ + C

Solution steps

\int \frac{x ^{3} + 3 x}{x ^{2} - 3 x - 10} d x

Long division

\frac{x ^{3} + 3 x}{x ^{2} - 3 x - 10} : x + 3 + \frac{22 x + 30}{x ^{2} - 3 x - 10}

= \int x + 3 + \frac{22 x + 30}{x ^{2} - 3 x - 10} 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 x d x + \int 3 d x + \int \frac{22 x + 30}{x ^{2} - 3 x - 10} d x

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

\int 3 d x = 3 x

\int \frac{22 x + 30}{x ^{2} - 3 x - 10} d x = 2 ln ∣ x + 2 ∣ + 20 ln ∣ x - 5 ∣

= \frac{x ^{2}}{2} + 3 x + 2 ln ∣ x + 2 ∣ + 20 ln ∣ x - 5 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} + 3 x + 2 ln ∣ x + 2 ∣ + 20 ln ∣ x - 5 ∣ + C

\int \frac{1}{x ^{2 - 4}} d x

\int x^{4} \cdot e^{(1 - x^{5})} d x

\int \frac{10}{9 x ^{2}} d x

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

What is the integral of (x^3+3x)/(x^2-3x-10) ?
The integral of (x^3+3x)/(x^2-3x-10) is (x^2)/2+3x+2ln|x+2|+20ln|x-5|+C