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

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

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

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

e^{2} - 1

Decimal

6.38905 \dots

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

= \frac{1}{2} \cdot \int_{0}^{4} e^{\frac{u}{2}} d u

Apply u-substitution

= \frac{1}{2} \cdot \int_{0}^{2} e^{v} \cdot 2 d v

Take the constant out:

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

= \frac{1}{2} \cdot 2 \cdot \int_{0}^{2} e^{v} d v

Use the common integral:

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

= \frac{1}{2} \cdot 2 [e^{v}]_{0}^{2}

Simplify

\frac{1}{2} \cdot 2 [e^{v}]_{0}^{2} : [e^{v}]_{0}^{2}

= [e^{v}]_{0}^{2}

Compute the boundaries:

e^{2} - 1

= e^{2} - 1

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