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

The integral from 0 to 2 of xe^{x/2} is 4

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

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

4

Solution steps

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

Apply u-substitution

= \int_{0}^{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_{0}^{1} e^{u} u d u

Apply Integration By Parts

= 4 [e^{u} u - \int e^{u} d u]_{0}^{1}

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

= 4 [e^{u} u - e^{u}]_{0}^{1}

Compute the boundaries:

1

= 4 \cdot 1

Simplify

= 4

What is the integral from 0 to 2 of xe^{x/2} ?
The integral from 0 to 2 of xe^{x/2} is 4