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

The integral from 1 to 2 of xe^{x/2} is 2sqrt(e)

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

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

2 e

Decimal

3.29744 \dots

Solution steps

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

Apply u-substitution

= \int_{\frac{1}{2}}^{1} 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_{\frac{1}{2}}^{1} e^{u} u d u

Apply Integration By Parts

= 4 [e^{u} u - \int e^{u} d u]_{\frac{1}{2}}^{1}

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

= 4 [e^{u} u - e^{u}]_{\frac{1}{2}}^{1}

Compute the boundaries:

\frac{e}{2}

= 4 \cdot \frac{e}{2}

Simplify

= 2 e

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