What is the integral of (x^3-5x-11)/(x^2-x-6) ?

The integral of (x^3-5x-11)/(x^2-x-6) is (x^2)/2+x+9/5 ln|x+2|+1/5 ln|x-3|+C

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integral of (x^3-5x-11)/(x^2-x-6)

\int \frac{x ^{3} - 5 x - 11}{x ^{2} - x - 6} d x

\frac{x ^{2}}{2} + x + \frac{9}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{x ^{3} - 5 x - 11}{x ^{2} - x - 6} d x

Take the partial fraction of

\frac{x ^{3} - 5 x - 11}{x ^{2} - x - 6} : x + 1 + \frac{9}{5 ( x + 2 )} + \frac{1}{5 ( x - 3 )}

= \int x + 1 + \frac{9}{5 ( x + 2 )} + \frac{1}{5 ( x - 3 )} 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 1 d x + \int \frac{9}{5 ( x + 2 )} d x + \int \frac{1}{5 ( x - 3 )} d x

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

\int 1 d x = x

\int \frac{9}{5 ( x + 2 )} d x = \frac{9}{5} ln ∣ x + 2 ∣

\int \frac{1}{5 ( x - 3 )} d x = \frac{1}{5} ln ∣ x - 3 ∣

= \frac{x ^{2}}{2} + x + \frac{9}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} + x + \frac{9}{5} ln ∣ x + 2 ∣ + \frac{1}{5} ln ∣ x - 3 ∣ + C

n = 1 \sum \infty \frac{5 n}{n ^{2} + 2}

\frac{d}{d x} (1 + 9 x^{4})

\frac{d}{d x} (x^{3} - \frac{1}{x ^{7}})

\frac{d y}{d x} - e^{x + y} = 0

x \to \infty lim ((\frac{( 4 + x )}{x})^{3 + \frac{x}{3}})

What is the integral of (x^3-5x-11)/(x^2-x-6) ?
The integral of (x^3-5x-11)/(x^2-x-6) is (x^2)/2+x+9/5 ln|x+2|+1/5 ln|x-3|+C