What is the integral of (x^2)/(4x+24) ?

The integral of (x^2)/(4x+24) is (x^2)/8-3/2 x+9ln|x+6|+C

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integral of (x^2)/(4x+24)

\int \frac{x ^{2}}{4 x + 24} d x

\frac{x ^{2}}{8} - \frac{3}{2} x + 9 ln ∣ x + 6 ∣ + C

Solution steps

\int \frac{x ^{2}}{4 x + 24} d x

Long division

\frac{x ^{2}}{4 x + 24} : \frac{x}{4} - \frac{3}{2} + \frac{9}{x + 6}

= \int \frac{x}{4} - \frac{3}{2} + \frac{9}{x + 6} 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{x}{4} d x - \int \frac{3}{2} d x + \int \frac{9}{x + 6} d x

\int \frac{x}{4} d x = \frac{x ^{2}}{8}

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

\int \frac{9}{x + 6} d x = 9 ln ∣ x + 6 ∣

= \frac{x ^{2}}{8} - \frac{3}{2} x + 9 ln ∣ x + 6 ∣

Add a constant to the solution

= \frac{x ^{2}}{8} - \frac{3}{2} x + 9 ln ∣ x + 6 ∣ + C

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What is the integral of (x^2)/(4x+24) ?
The integral of (x^2)/(4x+24) is (x^2)/8-3/2 x+9ln|x+6|+C