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

The integral from 2 to 4 of x(x^2-4)^2 is 288

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

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

288

Solution steps

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

Expand

x (x^{2} - 4)^{2} : x^{5} - 8 x^{3} + 16 x

= \int_{2}^{4} x^{5} - 8 x^{3} + 16 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}^{4} x^{5} d x - \int_{2}^{4} 8 x^{3} d x + \int_{2}^{4} 16 x d x

\int_{2}^{4} x^{5} d x = 672

\int_{2}^{4} 8 x^{3} d x = 480

\int_{2}^{4} 16 x d x = 96

= 672 - 480 + 96

Simplify

= 288

\int_{0}^{1} x sin (n x) d x

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What is the integral from 2 to 4 of x(x^2-4)^2 ?
The integral from 2 to 4 of x(x^2-4)^2 is 288