What is the integral from 5 to 6 of x(x-5)^2 ?

The integral from 5 to 6 of x(x-5)^2 is 23/12

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integral from 5 to 6 of x(x-5)^2

\int_{5}^{6} x (x - 5)^{2} d x

\frac{23}{12}

Decimal Notation

1.91666 \dots

Solution steps

\int_{5}^{6} x (x - 5)^{2} d x

Expand

x (x - 5)^{2} : x^{3} - 10 x^{2} + 25 x

= \int_{5}^{6} x^{3} - 10 x^{2} + 25 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_{5}^{6} x^{3} d x - \int_{5}^{6} 10 x^{2} d x + \int_{5}^{6} 25 x d x

\int_{5}^{6} x^{3} d x = \frac{671}{4}

\int_{5}^{6} 10 x^{2} d x = \frac{910}{3}

\int_{5}^{6} 25 x d x = \frac{275}{2}

= \frac{671}{4} - \frac{910}{3} + \frac{275}{2}

Simplify

\frac{671}{4} - \frac{910}{3} + \frac{275}{2} : \frac{23}{12}

= \frac{23}{12}

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What is the integral from 5 to 6 of x(x-5)^2 ?
The integral from 5 to 6 of x(x-5)^2 is 23/12