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

The integral of x(x-2)^2-x is (x^4)/4-(4x^3)/3+2x^2-(x^2)/2+C

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

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

\frac{x ^{4}}{4} - \frac{4 x ^{3}}{3} + 2 x^{2} - \frac{x ^{2}}{2} + C

Solution steps

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

\int x (x - 2)^{2} d x = \frac{x ^{4}}{4} - \frac{4 x ^{3}}{3} + 2 x^{2}

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

= \frac{x ^{4}}{4} - \frac{4 x ^{3}}{3} + 2 x^{2} - \frac{x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{4}}{4} - \frac{4 x ^{3}}{3} + 2 x^{2} - \frac{x ^{2}}{2} + C

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What is the integral of x(x-2)^2-x ?
The integral of x(x-2)^2-x is (x^4)/4-(4x^3)/3+2x^2-(x^2)/2+C