What is the integral of (-18x^2+12)/(x^3-2x-8) ?

The integral of (-18x^2+12)/(x^3-2x-8) is -6ln|x^3-2x-8|+C

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integral of (-18x^2+12)/(x^3-2x-8)

\int \frac{- 18 x ^{2} + 12}{x ^{3} - 2 x - 8} d x

- 6 ln x^{3} - 2 x - 8 + C

Solution steps

\int \frac{- 18 x ^{2} + 12}{x ^{3} - 2 x - 8} d x

Apply u-substitution

: \int - \frac{6}{u} d u

= \int - \frac{6}{u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 6 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 6 ln ∣ u ∣

Substitute back

u = x^{3} - 2 x - 8

= - 6 ln x^{3} - 2 x - 8

Add a constant to the solution

= - 6 ln x^{3} - 2 x - 8 + C

\int cot^{2} (6 x^{2}) x d x

\int \frac{1}{e ^{2 x}} d x

\int - 15 t^{\frac{2}{3}} d t

\int \frac{e ^{k x}}{k} d x

What is the integral of (-18x^2+12)/(x^3-2x-8) ?
The integral of (-18x^2+12)/(x^3-2x-8) is -6ln|x^3-2x-8|+C