What is the integral from 0 to 4 of xe^{(-1/2)x^2} ?

The integral from 0 to 4 of xe^{(-1/2)x^2} is 1-1/(e^8)

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integral from 0 to 4 of xe^{(-1/2)x^2}

\int_{0}^{4} x e^{(- \frac{1}{2}) x^{2}} d x

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

Decimal

0.99966 \dots

Solution steps

\int_{0}^{4} x e^{(- \frac{1}{2}) x^{2}} d x

Factor

x e^{(- \frac{1}{2}) x^{2}} : e^{- \frac{x ^{2}}{2}} x

= \int_{0}^{4} e^{- \frac{x ^{2}}{2}} x d x

Apply u-substitution

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

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

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

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

Take the constant out:

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

= - (- \int_{- 8}^{0} e^{u} d u)

Use the common integral:

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

= - (- [e^{u}]_{- 8}^{0})

Simplify

= [e^{u}]_{- 8}^{0}

Compute the boundaries:

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

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

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What is the integral from 0 to 4 of xe^{(-1/2)x^2} ?
The integral from 0 to 4 of xe^{(-1/2)x^2} is 1-1/(e^8)