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

The integral from 0 to 1 of xe^{x+2} is e^2

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

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

e^{2}

Decimal

7.38905 \dots

Solution steps

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

x e^{x + 2} = e^{2} x e^{x}

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

Take the constant out:

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

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

Apply Integration By Parts

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

= e^{2} [e^{x} x - \int e^{x} d x]_{0}^{1}

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

= e^{2} [e^{x} x - e^{x}]_{0}^{1}

Compute the boundaries:

1

= e^{2} \cdot 1

Simplify

= e^{2}

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