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

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

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

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

\frac{2}{3}

Decimal Notation

0.66666 \dots

Solution steps

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

Expand

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

= \int_{0}^{1} x^{2} + 2 x^{5} 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}^{1} x^{2} d x + \int_{0}^{1} 2 x^{5} d x

\int_{0}^{1} x^{2} d x = \frac{1}{3}

\int_{0}^{1} 2 x^{5} d x = \frac{1}{3}

= \frac{1}{3} + \frac{1}{3}

Simplify

\frac{1}{3} + \frac{1}{3} : \frac{2}{3}

= \frac{2}{3}

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