What is the integral from 2 to 10 of x(10-x)^2 ?

The integral from 2 to 10 of x(10-x)^2 is 2048/3

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integral from 2 to 10 of x(10-x)^2

\int_{2}^{10} x (10 - x)^{2} d x

\frac{2048}{3}

Decimal

682.66666 \dots

Solution steps

\int_{2}^{10} x (10 - x)^{2} d x

Expand

x (10 - x)^{2} : 100 x - 20 x^{2} + x^{3}

= \int_{2}^{10} 100 x - 20 x^{2} + x^{3} 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_{2}^{10} 100 x d x - \int_{2}^{10} 20 x^{2} d x + \int_{2}^{10} x^{3} d x

\int_{2}^{10} 100 x d x = 4800

\int_{2}^{10} 20 x^{2} d x = \frac{19840}{3}

\int_{2}^{10} x^{3} d x = 2496

= 4800 - \frac{19840}{3} + 2496

Simplify

4800 - \frac{19840}{3} + 2496 : \frac{2048}{3}

= \frac{2048}{3}

What is the integral from 2 to 10 of x(10-x)^2 ?
The integral from 2 to 10 of x(10-x)^2 is 2048/3