What is the integral of (x^3+5x^2+9x+7)/(x^2+5x+4) ?

The integral of (x^3+5x^2+9x+7)/(x^2+5x+4) is (x^2)/2+2/3 ln|x+1|+13/3 ln|x+4|+C

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integral of (x^3+5x^2+9x+7)/(x^2+5x+4)

\int \frac{x ^{3} + 5 x ^{2} + 9 x + 7}{x ^{2} + 5 x + 4} d x

\frac{x ^{2}}{2} + \frac{2}{3} ln ∣ x + 1 ∣ + \frac{13}{3} ln ∣ x + 4 ∣ + C

Solution steps

\int \frac{x ^{3} + 5 x ^{2} + 9 x + 7}{x ^{2} + 5 x + 4} d x

Long division

\frac{x ^{3} + 5 x ^{2} + 9 x + 7}{x ^{2} + 5 x + 4} : x + \frac{5 x + 7}{x ^{2} + 5 x + 4}

= \int x + \frac{5 x + 7}{x ^{2} + 5 x + 4} 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 \frac{5 x + 7}{x ^{2} + 5 x + 4} d x

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

\int \frac{5 x + 7}{x ^{2} + 5 x + 4} d x = \frac{2}{3} ln ∣ x + 1 ∣ + \frac{13}{3} ln ∣ x + 4 ∣

= \frac{x ^{2}}{2} + \frac{2}{3} ln ∣ x + 1 ∣ + \frac{13}{3} ln ∣ x + 4 ∣

Add a constant to the solution

= \frac{x ^{2}}{2} + \frac{2}{3} ln ∣ x + 1 ∣ + \frac{13}{3} ln ∣ x + 4 ∣ + C

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What is the integral of (x^3+5x^2+9x+7)/(x^2+5x+4) ?
The integral of (x^3+5x^2+9x+7)/(x^2+5x+4) is (x^2)/2+2/3 ln|x+1|+13/3 ln|x+4|+C