What is the integral of (x^4-4x^2-14x)/(x^2-2x-8) ?

The integral of (x^4-4x^2-14x)/(x^2-2x-8) is (x^3)/3+x^2+8x-14/3 ln|x+2|+68/3 ln|x-4|+4+C

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integral of (x^4-4x^2-14x)/(x^2-2x-8)

\int \frac{x ^{4} - 4 x ^{2} - 14 x}{x ^{2} - 2 x - 8} d x

\frac{x ^{3}}{3} + x^{2} + 8 x - \frac{14}{3} ln ∣ x + 2 ∣ + \frac{68}{3} ln ∣ x - 4 ∣ + 4 + C

Solution steps

\int \frac{x ^{4} - 4 x ^{2} - 14 x}{x ^{2} - 2 x - 8} d x

Expand

\frac{x ^{4} - 4 x ^{2} - 14 x}{x ^{2} - 2 x - 8} : \frac{x ^{4}}{x ^{2} - 2 x - 8} - \frac{4 x ^{2}}{x ^{2} - 2 x - 8} - \frac{14 x}{x ^{2} - 2 x - 8}

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{x ^{4}}{x ^{2} - 2 x - 8} d x - \int \frac{4 x ^{2}}{x ^{2} - 2 x - 8} d x - \int \frac{14 x}{x ^{2} - 2 x - 8} d x

\int \frac{x ^{4}}{x ^{2} - 2 x - 8} d x = \frac{x ^{3}}{3} + x^{2} + 12 x - \frac{8}{3} ln ∣ x + 2 ∣ + \frac{128}{3} ln ∣ x - 4 ∣

\int \frac{4 x ^{2}}{x ^{2} - 2 x - 8} d x = 4 (x - 1 - \frac{2}{3} ln ∣ x + 2 ∣ + \frac{8}{3} ln ∣ x - 4 ∣)

\int \frac{14 x}{x ^{2} - 2 x - 8} d x = 14 (\frac{1}{3} ln ∣ x + 2 ∣ + \frac{2}{3} ln ∣ x - 4 ∣)

= \frac{x ^{3}}{3} + x^{2} + 12 x - \frac{8}{3} ln ∣ x + 2 ∣ + \frac{128}{3} ln ∣ x - 4 ∣ - 4 (x - 1 - \frac{2}{3} ln ∣ x + 2 ∣ + \frac{8}{3} ln ∣ x - 4 ∣) - 14 (\frac{1}{3} ln ∣ x + 2 ∣ + \frac{2}{3} ln ∣ x - 4 ∣)

Simplify

\frac{x ^{3}}{3} + x^{2} + 12 x - \frac{8}{3} ln ∣ x + 2 ∣ + \frac{128}{3} ln ∣ x - 4 ∣ - 4 (x - 1 - \frac{2}{3} ln ∣ x + 2 ∣ + \frac{8}{3} ln ∣ x - 4 ∣) - 14 (\frac{1}{3} ln ∣ x + 2 ∣ + \frac{2}{3} ln ∣ x - 4 ∣) : \frac{x ^{3}}{3} + x^{2} + 8 x - \frac{14}{3} ln ∣ x + 2 ∣ + \frac{68}{3} ln ∣ x - 4 ∣ + 4

= \frac{x ^{3}}{3} + x^{2} + 8 x - \frac{14}{3} ln ∣ x + 2 ∣ + \frac{68}{3} ln ∣ x - 4 ∣ + 4

Add a constant to the solution

= \frac{x ^{3}}{3} + x^{2} + 8 x - \frac{14}{3} ln ∣ x + 2 ∣ + \frac{68}{3} ln ∣ x - 4 ∣ + 4 + C

\int \frac{e ^{2 x} + 2 e ^{3 x}}{4 e ^{2 x}} d x

\int \frac{3}{x ^{3} - 8} d x

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

\int \frac{5}{x ^{\frac{3}{2}} - 25 x} d x

\int 7 - 4 x d x

What is the integral of (x^4-4x^2-14x)/(x^2-2x-8) ?
The integral of (x^4-4x^2-14x)/(x^2-2x-8) is (x^3)/3+x^2+8x-14/3 ln|x+2|+68/3 ln|x-4|+4+C