What is the integral of (8x^3-12x^2+7x+2)/(2x-2) ?

The integral of (8x^3-12x^2+7x+2)/(2x-2) is (4x^3)/3-x^2+3/2 x+5/2 ln|2x-2|+C

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integral of (8x^3-12x^2+7x+2)/(2x-2)

\int \frac{8 x ^{3} - 12 x ^{2} + 7 x + 2}{2 x - 2} d x

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

Solution steps

\int \frac{8 x ^{3} - 12 x ^{2} + 7 x + 2}{2 x - 2} d x

Long division

\frac{8 x ^{3} - 12 x ^{2} + 7 x + 2}{2 x - 2} : 4 x^{2} - 2 x + \frac{3}{2} + \frac{5}{2 x - 2}

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

\int 4 x^{2} d x = \frac{4 x ^{3}}{3}

\int 2 x d x = x^{2}

\int \frac{3}{2} d x = \frac{3}{2} x

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

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

Add a constant to the solution

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

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What is the integral of (8x^3-12x^2+7x+2)/(2x-2) ?
The integral of (8x^3-12x^2+7x+2)/(2x-2) is (4x^3)/3-x^2+3/2 x+5/2 ln|2x-2|+C