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

The integral of (x^3)/(3+x) is (x^3)/3-(3x^2)/2+9x-27ln|x+3|+C

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

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

\frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 9 x - 27 ln ∣ x + 3 ∣ + C

Solution steps

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

Long division

\frac{x ^{3}}{3 + x} : x^{2} - 3 x + 9 - \frac{27}{x + 3}

= \int x^{2} - 3 x + 9 - \frac{27}{x + 3} 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^{2} d x - \int 3 x d x + \int 9 d x - \int \frac{27}{x + 3} d x

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

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

\int 9 d x = 9 x

\int \frac{27}{x + 3} d x = 27 ln ∣ x + 3 ∣

= \frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 9 x - 27 ln ∣ x + 3 ∣

Add a constant to the solution

= \frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 9 x - 27 ln ∣ x + 3 ∣ + C

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

\frac{d}{d x} ((ln (x))^{9})

\int 9 e^{x} d x

\int \frac{9 x ^{2} + x + 9}{( x ^{2} + 1 ) ^{2}} d x

f (x) = \frac{x + 2}{x - 1}

What is the integral of (x^3)/(3+x) ?
The integral of (x^3)/(3+x) is (x^3)/3-(3x^2)/2+9x-27ln|x+3|+C