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

The integral from 0 to 1 of 2/5 x^2(x+2) is 11/30

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integral from 0 to 1 of 2/5 x^2(x+2)

\int_{0}^{1} \frac{2}{5} x^{2} (x + 2) d x

\frac{11}{30}

Decimal

0.36666 \dots

Solution steps

\int_{0}^{1} \frac{2}{5} x^{2} (x + 2) d x

Take the constant out:

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

= \frac{2}{5} \cdot \int_{0}^{1} x^{2} (x + 2) d x

Apply Integration By Parts

: [\frac{1}{3} x^{3} (x + 2) - \int \frac{x ^{3}}{3} d x]_{0}^{1}

= \frac{2}{5} [\frac{1}{3} x^{3} (x + 2) - \int \frac{x ^{3}}{3} d x]_{0}^{1}

\int \frac{x ^{3}}{3} d x = \frac{x ^{4}}{12}

= \frac{2}{5} [\frac{1}{3} x^{3} (x + 2) - \frac{x ^{4}}{12}]_{0}^{1}

Simplify

\frac{2}{5} [\frac{1}{3} x^{3} (x + 2) - \frac{x ^{4}}{12}]_{0}^{1} : \frac{2}{5} [\frac{3 x ^{4} + 8 x ^{3}}{12}]_{0}^{1}

= \frac{2}{5} [\frac{3 x ^{4} + 8 x ^{3}}{12}]_{0}^{1}

Compute the boundaries:

\frac{11}{12}

= \frac{2}{5} \cdot \frac{11}{12}

Simplify

= \frac{11}{30}

\int \frac{1}{e ^{- 4 x}} d x

x \to 2 lim (\frac{x - 5}{x - 2})

x \to 0 lim (\frac{z ^{x} - 1}{x})

x \to 0 lim (x + 4)

\frac{d}{d x} ((x^{- 2} + x^{- 3}) (x^{5} - 9 x^{2}))

What is the integral from 0 to 1 of 2/5 x^2(x+2) ?
The integral from 0 to 1 of 2/5 x^2(x+2) is 11/30