What is the integral from 0 to 1 of x^2e^{-11x} ?

The integral from 0 to 1 of x^2e^{-11x} is (2e^{11}-145)/(1331e^{11)}

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integral from 0 to 1 of x^2e^{-11x}

\int_{0}^{1} x^{2} e^{- 11 x} d x

\frac{2 e ^{11} - 145}{1331 e ^{11}}

Decimal Notation

0.00150 \dots

Solution steps

\int_{0}^{1} x^{2} e^{- 11 x} d x

Apply u-substitution

: \int_{0}^{- 11} - \frac{e ^{u} u ^{2}}{1331} d u

= \int_{0}^{- 11} - \frac{e ^{u} u ^{2}}{1331} d u

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

= - \int_{- 11}^{0} - \frac{e ^{u} u ^{2}}{1331} d u

Take the constant out:

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

= - (- \frac{1}{1331} \cdot \int_{- 11}^{0} e^{u} u^{2} d u)

Apply Integration By Parts

: [u^{2} e^{u} - \int 2 u e^{u} d u]_{- 11}^{0}

= - (- \frac{1}{1331} [u^{2} e^{u} - \int 2 u e^{u} d u]_{- 11}^{0})

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

= - (- \frac{1}{1331} [u^{2} e^{u} - 2 (e^{u} u - e^{u})]_{- 11}^{0})

Simplify

= \frac{1}{1331} [e^{u} u^{2} - 2 (e^{u} u - e^{u})]_{- 11}^{0}

Compute the boundaries:

2 - \frac{145}{e ^{11}}

= \frac{1}{1331} (2 - \frac{145}{e ^{11}})

Simplify

= \frac{2 e ^{11} - 145}{1331 e ^{11}}

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What is the integral from 0 to 1 of x^2e^{-11x} ?
The integral from 0 to 1 of x^2e^{-11x} is (2e^{11}-145)/(1331e^{11)}