What is the integral of ((7x^2+5x-6))/((x^2-4)(x+2)) ?

The integral of ((7x^2+5x-6))/((x^2-4)(x+2)) is 5ln|x+2|+3/(x+2)+2ln|x-2|+C

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integral of ((7x^2+5x-6))/((x^2-4)(x+2))

\int \frac{( 7 x ^{2} + 5 x - 6 )}{( x ^{2} - 4 ) ( x + 2 )} d x

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

Solution steps

\int \frac{7 x ^{2} + 5 x - 6}{( x ^{2} - 4 ) ( x + 2 )} d x

Take the partial fraction of

\frac{7 x ^{2} + 5 x - 6}{( x ^{2} - 4 ) ( x + 2 )} : \frac{5}{x + 2} - \frac{3}{( x + 2 ) ^{2}} + \frac{2}{x - 2}

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

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

\int \frac{3}{( x + 2 ) ^{2}} d x = - \frac{3}{x + 2}

\int \frac{2}{x - 2} d x = 2 ln ∣ x - 2 ∣

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

Simplify

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

Add a constant to the solution

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

What is the integral of ((7x^2+5x-6))/((x^2-4)(x+2)) ?
The integral of ((7x^2+5x-6))/((x^2-4)(x+2)) is 5ln|x+2|+3/(x+2)+2ln|x-2|+C