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

The integral from 0 to 3 of xe^{x/2} is 4((e^{3/2})/2+1)

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integral from 0 to 3 of xe^{x/2}

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

4 (\frac{e ^{\frac{3}{2}}}{2} + 1)

Decimal Notation

12.96337 \dots

Solution steps

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

Apply u-substitution

: \int_{0}^{\frac{3}{2}} 4 e^{u} u d u

= \int_{0}^{\frac{3}{2}} 4 e^{u} u d u

Take the constant out:

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

= 4 \cdot \int_{0}^{\frac{3}{2}} e^{u} u d u

Apply Integration By Parts

: [e^{u} u - \int e^{u} d u]_{0}^{\frac{3}{2}}

= 4 [e^{u} u - \int e^{u} d u]_{0}^{\frac{3}{2}}

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

= 4 [e^{u} u - e^{u}]_{0}^{\frac{3}{2}}

Compute the boundaries:

\frac{e ^{\frac{3}{2}}}{2} + 1

= 4 (\frac{e ^{\frac{3}{2}}}{2} + 1)

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What is the integral from 0 to 3 of xe^{x/2} ?
The integral from 0 to 3 of xe^{x/2} is 4((e^{3/2})/2+1)