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

The integral from 0 to 1/3 of x^2e^{-3x} is (2e-5)/(27e)

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integral from 0 to 1/3 of x^2e^{-3x}

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

\frac{2 e - 5}{27 e}

Decimal

0.00594 \dots

Solution steps

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

Apply u-substitution

: \int_{0}^{- 1} - \frac{e ^{u} u ^{2}}{27} d u

= \int_{0}^{- 1} - \frac{e ^{u} u ^{2}}{27} d u

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

= - \int_{- 1}^{0} - \frac{e ^{u} u ^{2}}{27} d u

Take the constant out:

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

= - (- \frac{1}{27} \cdot \int_{- 1}^{0} e^{u} u^{2} d u)

Apply Integration By Parts

: [u^{2} e^{u} - \int 2 u e^{u} d u]_{- 1}^{0}

= - (- \frac{1}{27} [u^{2} e^{u} - \int 2 u e^{u} d u]_{- 1}^{0})

\int 2 u e^{u} d u = 2 (e^{u} u - e^{u})

= - (- \frac{1}{27} [u^{2} e^{u} - 2 (e^{u} u - e^{u})]_{- 1}^{0})

Simplify

= \frac{1}{27} [e^{u} u^{2} - 2 (e^{u} u - e^{u})]_{- 1}^{0}

Compute the boundaries:

2 - \frac{5}{e}

= \frac{1}{27} (2 - \frac{5}{e})

Simplify

= \frac{2 e - 5}{27 e}

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What is the integral from 0 to 1/3 of x^2e^{-3x} ?
The integral from 0 to 1/3 of x^2e^{-3x} is (2e-5)/(27e)