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

The integral of (x^2)/(x^2-x-6) is x-1/2-4/5 ln|2x+4|+9/5 ln|2x-6|+C

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integral of (x^2)/(x^2-x-6)

\int \frac{x ^{2}}{x ^{2} - x - 6} d x

x - \frac{1}{2} - \frac{4}{5} ln ∣ 2 x + 4 ∣ + \frac{9}{5} ln ∣ 2 x - 6 ∣ + C

Solution steps

\int \frac{x ^{2}}{x ^{2} - x - 6} d x

Complete the square

x^{2} - x - 6 : (x - \frac{1}{2})^{2} - \frac{25}{4}

= \int \frac{x ^{2}}{( x - \frac{1}{2} ) ^{2} - \frac{25}{4}} d x

Apply u-substitution

: \int \frac{( 2 u + 1 ) ^{2}}{4 u ^{2} - 25} d u

= \int \frac{( 2 u + 1 ) ^{2}}{4 u ^{2} - 25} d u

Take the partial fraction of

\frac{( 2 u + 1 ) ^{2}}{4 u ^{2} - 25} : 1 - \frac{8}{5 ( 2 u + 5 )} + \frac{18}{5 ( 2 u - 5 )}

= \int 1 - \frac{8}{5 ( 2 u + 5 )} + \frac{18}{5 ( 2 u - 5 )} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u - \int \frac{8}{5 ( 2 u + 5 )} d u + \int \frac{18}{5 ( 2 u - 5 )} d u

\int 1 d u = u

\int \frac{8}{5 ( 2 u + 5 )} d u = \frac{4}{5} ln ∣ 2 u + 5 ∣

\int \frac{18}{5 ( 2 u - 5 )} d u = \frac{9}{5} ln ∣ 2 u - 5 ∣

= u - \frac{4}{5} ln ∣ 2 u + 5 ∣ + \frac{9}{5} ln ∣ 2 u - 5 ∣

Substitute back

u = x - \frac{1}{2}

= x - \frac{1}{2} - \frac{4}{5} ln 2 (x - \frac{1}{2}) + 5 + \frac{9}{5} ln 2 (x - \frac{1}{2}) - 5

Simplify

x - \frac{1}{2} - \frac{4}{5} ln 2 (x - \frac{1}{2}) + 5 + \frac{9}{5} ln 2 (x - \frac{1}{2}) - 5 : x - \frac{1}{2} - \frac{4}{5} ln ∣ 2 x + 4 ∣ + \frac{9}{5} ln ∣ 2 x - 6 ∣

= x - \frac{1}{2} - \frac{4}{5} ln ∣ 2 x + 4 ∣ + \frac{9}{5} ln ∣ 2 x - 6 ∣

Add a constant to the solution

= x - \frac{1}{2} - \frac{4}{5} ln ∣ 2 x + 4 ∣ + \frac{9}{5} ln ∣ 2 x - 6 ∣ + C

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What is the integral of (x^2)/(x^2-x-6) ?
The integral of (x^2)/(x^2-x-6) is x-1/2-4/5 ln|2x+4|+9/5 ln|2x-6|+C