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

The integral from 0 to 1/2 of xe^{3x} is (e^{3/2}+2)/(18)

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

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

\frac{e ^{\frac{3}{2}} + 2}{18}

Decimal

0.36009 \dots

Solution steps

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

Apply u-substitution

: \int_{0}^{\frac{3}{2}} \frac{e ^{u} u}{9} d u

= \int_{0}^{\frac{3}{2}} \frac{e ^{u} u}{9} d u

Take the constant out:

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

= \frac{1}{9} \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}}

= \frac{1}{9} [e^{u} u - \int e^{u} d u]_{0}^{\frac{3}{2}}

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

= \frac{1}{9} [e^{u} u - e^{u}]_{0}^{\frac{3}{2}}

Compute the boundaries:

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

= \frac{1}{9} (\frac{e ^{\frac{3}{2}}}{2} + 1)

Simplify

= \frac{e ^{\frac{3}{2}} + 2}{18}

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What is the integral from 0 to 1/2 of xe^{3x} ?
The integral from 0 to 1/2 of xe^{3x} is (e^{3/2}+2)/(18)