What is the integral from 1 to 2 of x(x-1)^2 ?

The integral from 1 to 2 of x(x-1)^2 is 7/12

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integral from 1 to 2 of x(x-1)^2

\int_{1}^{2} x (x - 1)^{2} d x

\frac{7}{12}

Decimal Notation

0.58333 \dots

Solution steps

\int_{1}^{2} x (x - 1)^{2} d x

Expand

x (x - 1)^{2} : x^{3} - 2 x^{2} + x

= \int_{1}^{2} x^{3} - 2 x^{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_{1}^{2} x^{3} d x - \int_{1}^{2} 2 x^{2} d x + \int_{1}^{2} x d x

\int_{1}^{2} x^{3} d x = \frac{15}{4}

\int_{1}^{2} 2 x^{2} d x = \frac{14}{3}

\int_{1}^{2} x d x = \frac{3}{2}

= \frac{15}{4} - \frac{14}{3} + \frac{3}{2}

Simplify

\frac{15}{4} - \frac{14}{3} + \frac{3}{2} : \frac{7}{12}

= \frac{7}{12}

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What is the integral from 1 to 2 of x(x-1)^2 ?
The integral from 1 to 2 of x(x-1)^2 is 7/12