What is the integral of (6x^2+x+27)/(8x^2-20x-12) ?

The integral of (6x^2+x+27)/(8x^2-20x-12) is 3/4 x-ln|2x+1|+3ln|x-3|+C

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integral of (6x^2+x+27)/(8x^2-20x-12)

\int \frac{6 x ^{2} + x + 27}{8 x ^{2} - 20 x - 12} d x

\frac{3}{4} x - ln ∣ 2 x + 1 ∣ + 3 ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{6 x ^{2} + x + 27}{8 x ^{2} - 20 x - 12} d x

Long division

\frac{6 x ^{2} + x + 27}{8 x ^{2} - 20 x - 12} : \frac{3}{4} + \frac{4 x + 9}{2 x ^{2} - 5 x - 3}

= \int \frac{3}{4} + \frac{4 x + 9}{2 x ^{2} - 5 x - 3} 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{3}{4} d x + \int \frac{4 x + 9}{2 x ^{2} - 5 x - 3} d x

\int \frac{3}{4} d x = \frac{3}{4} x

\int \frac{4 x + 9}{2 x ^{2} - 5 x - 3} d x = - ln ∣ 2 x + 1 ∣ + 3 ln ∣ x - 3 ∣

= \frac{3}{4} x - ln ∣ 2 x + 1 ∣ + 3 ln ∣ x - 3 ∣

Add a constant to the solution

= \frac{3}{4} x - ln ∣ 2 x + 1 ∣ + 3 ln ∣ x - 3 ∣ + C

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What is the integral of (6x^2+x+27)/(8x^2-20x-12) ?
The integral of (6x^2+x+27)/(8x^2-20x-12) is 3/4 x-ln|2x+1|+3ln|x-3|+C