What is the integral from 0 to 4 of x(x^2+2)^2 ?

The integral from 0 to 4 of x(x^2+2)^2 is 2912/3

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integral from 0 to 4 of x(x^2+2)^2

\int_{0}^{4} x (x^{2} + 2)^{2} d x

\frac{2912}{3}

Solution steps

\int_{0}^{4} x (x^{2} + 2)^{2} d x

Expand

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

= \int_{0}^{4} x^{5} + 4 x^{3} + 4 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_{0}^{4} x^{5} d x + \int_{0}^{4} 4 x^{3} d x + \int_{0}^{4} 4 x d x

\int_{0}^{4} x^{5} d x = \frac{2048}{3}

\int_{0}^{4} 4 x^{3} d x = 256

\int_{0}^{4} 4 x d x = 32

= \frac{2048}{3} + 256 + 32

Simplify

\frac{2048}{3} + 256 + 32 : \frac{2912}{3}

= \frac{2912}{3}

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\int_{0}^{0.4} ln (1 + x^{4}) d x

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What is the integral from 0 to 4 of x(x^2+2)^2 ?
The integral from 0 to 4 of x(x^2+2)^2 is 2912/3