What is the integral of (x^3+x^2)/(x^2+7x+10) ?

The integral of (x^3+x^2)/(x^2+7x+10) is (x^2)/2-6x-4/3 ln|x+2|+100/3 ln|x+5|+C

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integral of (x^3+x^2)/(x^2+7x+10)

\int \frac{x ^{3} + x ^{2}}{x ^{2} + 7 x + 10} d x

\frac{x ^{2}}{2} - 6 x - \frac{4}{3} ln ∣ x + 2 ∣ + \frac{100}{3} ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{x ^{3} + x ^{2}}{x ^{2} + 7 x + 10} d x

Long division

\frac{x ^{3} + x ^{2}}{x ^{2} + 7 x + 10} : x - 6 + \frac{32 x + 60}{x ^{2} + 7 x + 10}

= \int x - 6 + \frac{32 x + 60}{x ^{2} + 7 x + 10} 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 6 d x + \int \frac{32 x + 60}{x ^{2} + 7 x + 10} d x

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

\int 6 d x = 6 x

\int \frac{32 x + 60}{x ^{2} + 7 x + 10} d x = - \frac{4}{3} ln ∣ x + 2 ∣ + \frac{100}{3} ln ∣ x + 5 ∣

= \frac{x ^{2}}{2} - 6 x - \frac{4}{3} ln ∣ x + 2 ∣ + \frac{100}{3} ln ∣ x + 5 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} - 6 x - \frac{4}{3} ln ∣ x + 2 ∣ + \frac{100}{3} ln ∣ x + 5 ∣ + C

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What is the integral of (x^3+x^2)/(x^2+7x+10) ?
The integral of (x^3+x^2)/(x^2+7x+10) is (x^2)/2-6x-4/3 ln|x+2|+100/3 ln|x+5|+C