What is the integral from 0 to 1 of 2/7 x^2(x+3) ?

The integral from 0 to 1 of 2/7 x^2(x+3) is 5/14

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integral from 0 to 1 of 2/7 x^2(x+3)

\int_{0}^{1} \frac{2}{7} x^{2} (x + 3) d x

\frac{5}{14}

Decimal Notation

0.35714 \dots

Solution steps

\int_{0}^{1} \frac{2}{7} x^{2} (x + 3) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{2}{7} \cdot \int_{0}^{1} x^{2} (x + 3) d x

Expand

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

= \frac{2}{7} \cdot \int_{0}^{1} x^{3} + 3 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

= \frac{2}{7} (\int_{0}^{1} x^{3} d x + \int_{0}^{1} 3 x^{2} d x)

\int_{0}^{1} x^{3} d x = \frac{1}{4}

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

= \frac{2}{7} (\frac{1}{4} + 1)

Simplify

\frac{2}{7} (\frac{1}{4} + 1) : \frac{5}{14}

= \frac{5}{14}

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What is the integral from 0 to 1 of 2/7 x^2(x+3) ?
The integral from 0 to 1 of 2/7 x^2(x+3) is 5/14