What is the integral from 0 to 2 of x^2(x^3+1) ?

The integral from 0 to 2 of x^2(x^3+1) is 40/3

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integral from 0 to 2 of x^2(x^3+1)

\int_{0}^{2} x^{2} (x^{3} + 1) d x

\frac{40}{3}

Decimal

13.33333 \dots

Solution steps

\int_{0}^{2} x^{2} (x^{3} + 1) d x

Expand

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

= \int_{0}^{2} x^{5} + x^{2} 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}^{2} x^{5} d x + \int_{0}^{2} x^{2} d x

\int_{0}^{2} x^{5} d x = \frac{32}{3}

\int_{0}^{2} x^{2} d x = \frac{8}{3}

= \frac{32}{3} + \frac{8}{3}

Simplify

\frac{32}{3} + \frac{8}{3} : \frac{40}{3}

= \frac{40}{3}

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What is the integral from 0 to 2 of x^2(x^3+1) ?
The integral from 0 to 2 of x^2(x^3+1) is 40/3