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

The integral of ((3x^2-2))/((x^2-2x-8)) is 3x-5/3 ln|x+2|+23/3 ln|x-4|+C

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integral of ((3x^2-2))/((x^2-2x-8))

integral

\frac{( 3 x ^{2} - 2 )}{( x ^{2} - 2 x - 8 )}

3 x - \frac{5}{3} ln ∣ x + 2 ∣ + \frac{23}{3} ln ∣ x - 4 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{3 x ^{2} - 2}{x ^{2} - 2 x - 8} : 3 - \frac{5}{3 ( x + 2 )} + \frac{23}{3 ( x - 4 )}

= \int 3 - \frac{5}{3 ( x + 2 )} + \frac{23}{3 ( x - 4 )} 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 3 d x - \int \frac{5}{3 ( x + 2 )} d x + \int \frac{23}{3 ( x - 4 )} d x

\int 3 d x = 3 x

\int \frac{5}{3 ( x + 2 )} d x = \frac{5}{3} ln ∣ x + 2 ∣

\int \frac{23}{3 ( x - 4 )} d x = \frac{23}{3} ln ∣ x - 4 ∣

= 3 x - \frac{5}{3} ln ∣ x + 2 ∣ + \frac{23}{3} ln ∣ x - 4 ∣

Add a constant to the solution

= 3 x - \frac{5}{3} ln ∣ x + 2 ∣ + \frac{23}{3} ln ∣ x - 4 ∣ + C

\frac{1}{( tan ( x ) + cot ( x ) ) ^{2}}

\frac{1}{x ^{\frac{1}{2}}}

y = x^{2}

\frac{x}{( 1 + x ^{3} )}

\frac{e ^{x}}{( e ^{x} + 1 ) ^{2}}

What is the integral of ((3x^2-2))/((x^2-2x-8)) ?
The integral of ((3x^2-2))/((x^2-2x-8)) is 3x-5/3 ln|x+2|+23/3 ln|x-4|+C