What is the integral of (x^3+27)/(x+3) ?

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

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integral of (x^3+27)/(x+3)

\int \frac{x ^{3} + 27}{x + 3} d x

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

Solution steps

\int \frac{x ^{3} + 27}{x + 3} d x

Expand

\frac{x ^{3} + 27}{x + 3} : \frac{x ^{3}}{x + 3} + \frac{27}{x + 3}

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 ^{3}}{x + 3} d x + \int \frac{27}{x + 3} d x

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

\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 ∣ + 27 ln ∣ x + 3 ∣

Simplify

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

Add a constant to the solution

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

\int fdf

\int_{\frac{π}{4}}^{π} sin (x) d x

\frac{d}{d x} (cos^{4} (x) + sec (x^{3}))

\int cos (x) d x

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