What is the integral from 2 to 5 of x(x-3)^4 ?

The integral from 2 to 5 of x(x-3)^4 is 303/10

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integral from 2 to 5 of x(x-3)^4

\int_{2}^{5} x (x - 3)^{4} d x

\frac{303}{10}

Decimal

30.3

Solution steps

\int_{2}^{5} x (x - 3)^{4} d x

Expand

x (x - 3)^{4} : x^{5} - 12 x^{4} + 54 x^{3} - 108 x^{2} + 81 x

= \int_{2}^{5} x^{5} - 12 x^{4} + 54 x^{3} - 108 x^{2} + 81 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_{2}^{5} x^{5} d x - \int_{2}^{5} 12 x^{4} d x + \int_{2}^{5} 54 x^{3} d x - \int_{2}^{5} 108 x^{2} d x + \int_{2}^{5} 81 x d x

\int_{2}^{5} x^{5} d x = \frac{5187}{2}

\int_{2}^{5} 12 x^{4} d x = \frac{37116}{5}

\int_{2}^{5} 54 x^{3} d x = \frac{16443}{2}

\int_{2}^{5} 108 x^{2} d x = 4212

\int_{2}^{5} 81 x d x = \frac{1701}{2}

= \frac{5187}{2} - \frac{37116}{5} + \frac{16443}{2} - 4212 + \frac{1701}{2}

Simplify

\frac{5187}{2} - \frac{37116}{5} + \frac{16443}{2} - 4212 + \frac{1701}{2} : \frac{303}{10}

= \frac{303}{10}

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\int_{0}^{2} ln (x^{3}) d x

\int_{3 x}^{8} \frac{x + y}{60} d y

What is the integral from 2 to 5 of x(x-3)^4 ?
The integral from 2 to 5 of x(x-3)^4 is 303/10