What is the integral of ((x-2)^3)/x ?

The integral of ((x-2)^3)/x is (x^3)/3-3x^2+12x-8ln|x|+C

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integral of ((x-2)^3)/x

\int \frac{( x - 2 ) ^{3}}{x} d x

\frac{x ^{3}}{3} - 3 x^{2} + 12 x - 8 ln ∣ x ∣ + C

Solution steps

\int \frac{( x - 2 ) ^{3}}{x} d x

Expand

\frac{( x - 2 ) ^{3}}{x} : x^{2} - 6 x + 12 - \frac{8}{x}

= \int x^{2} - 6 x + 12 - \frac{8}{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 6 x d x + \int 12 d x - \int \frac{8}{x} d x

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

\int 6 x d x = 3 x^{2}

\int 12 d x = 12 x

\int \frac{8}{x} d x = 8 ln ∣ x ∣

= \frac{x ^{3}}{3} - 3 x^{2} + 12 x - 8 ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{3}}{3} - 3 x^{2} + 12 x - 8 ln ∣ x ∣ + C

What is the integral of ((x-2)^3)/x ?
The integral of ((x-2)^3)/x is (x^3)/3-3x^2+12x-8ln|x|+C