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

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

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

\int (\frac{( x - 3 ) ^{3}}{x}) d x

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

Solution steps

\int \frac{( x - 3 ) ^{3}}{x} d x

Long division

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

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

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

\int 9 x d x = \frac{9 x ^{2}}{2}

\int 27 d x = 27 x

\int \frac{27}{x} d x = 27 ln ∣ x ∣

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

Add a constant to the solution

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

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