What is the integral from 0 to 2 of x^2e^{-x^3} ?

The integral from 0 to 2 of x^2e^{-x^3} is (e^8-1)/(3e^8)

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integral from 0 to 2 of x^2e^{-x^3}

\int_{0}^{2} x^{2} e^{- x^{3}} d x

\frac{e ^{8} - 1}{3 e ^{8}}

Decimal

0.33322 \dots

Solution steps

\int_{0}^{2} x^{2} e^{- x^{3}} d x

Apply u-substitution

: \int_{0}^{- 8} - \frac{e ^{u}}{3} d u

= \int_{0}^{- 8} - \frac{e ^{u}}{3} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{- 8}^{0} - \frac{e ^{u}}{3} d u

Take the constant out:

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

= - (- \frac{1}{3} \cdot \int_{- 8}^{0} e^{u} d u)

Use the common integral:

\int e^{u} d u = e^{u}

= - (- \frac{1}{3} [e^{u}]_{- 8}^{0})

Simplify

= \frac{1}{3} [e^{u}]_{- 8}^{0}

Compute the boundaries:

1 - \frac{1}{e ^{8}}

= \frac{1}{3} (1 - \frac{1}{e ^{8}})

Simplify

= \frac{e ^{8} - 1}{3 e ^{8}}

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What is the integral from 0 to 2 of x^2e^{-x^3} ?
The integral from 0 to 2 of x^2e^{-x^3} is (e^8-1)/(3e^8)