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

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

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

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

\frac{x ^{2}}{2} + 4 x - 2 ln ∣ x - 1 ∣ + 12 ln ∣ x - 2 ∣ + C

Solution steps

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

Long division

\frac{x ^{3} + x ^{2}}{x ^{2} - 3 x + 2} : x + 4 + \frac{10 x - 8}{x ^{2} - 3 x + 2}

= \int x + 4 + \frac{10 x - 8}{x ^{2} - 3 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 x d x + \int 4 d x + \int \frac{10 x - 8}{x ^{2} - 3 x + 2} d x

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

\int 4 d x = 4 x

\int \frac{10 x - 8}{x ^{2} - 3 x + 2} d x = - 2 ln ∣ x - 1 ∣ + 12 ln ∣ x - 2 ∣

= \frac{x ^{2}}{2} + 4 x - 2 ln ∣ x - 1 ∣ + 12 ln ∣ x - 2 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} + 4 x - 2 ln ∣ x - 1 ∣ + 12 ln ∣ x - 2 ∣ + C

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