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

The integral from 0 to 1/2 of xe^{2x} is 1/4

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

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

\frac{1}{4}

Decimal Notation

0.25

Solution steps

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

Apply u-substitution

: \int_{0}^{1} \frac{e ^{u} u}{4} d u

= \int_{0}^{1} \frac{e ^{u} u}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int_{0}^{1} e^{u} u d u

Apply Integration By Parts

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

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

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

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

Compute the boundaries:

1

= \frac{1}{4} \cdot 1

Simplify

= \frac{1}{4}

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