What is the integral from 1 to 5 of x(x^2-1)^2 ?

The integral from 1 to 5 of x(x^2-1)^2 is 2304

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integral from 1 to 5 of x(x^2-1)^2

\int_{1}^{5} x (x^{2} - 1)^{2} d x

2304

Solution steps

\int_{1}^{5} x (x^{2} - 1)^{2} d x

Expand

x (x^{2} - 1)^{2} : x^{5} - 2 x^{3} + x

= \int_{1}^{5} x^{5} - 2 x^{3} + 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}^{5} x^{5} d x - \int_{1}^{5} 2 x^{3} d x + \int_{1}^{5} x d x

\int_{1}^{5} x^{5} d x = 2604

\int_{1}^{5} 2 x^{3} d x = 312

\int_{1}^{5} x d x = 12

= 2604 - 312 + 12

Simplify

= 2304

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What is the integral from 1 to 5 of x(x^2-1)^2 ?
The integral from 1 to 5 of x(x^2-1)^2 is 2304