What is the integral from 0 to 3 of xe^{3x} ?

The integral from 0 to 3 of xe^{3x} is (8e^9+1)/9

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integral from 0 to 3 of xe^{3x}

\int_{0}^{3} x e^{3 x} d x

\frac{8 e ^{9} + 1}{9}

Decimal

7202.85238 \dots

Solution steps

\int_{0}^{3} x e^{3 x} d x

Apply u-substitution

: \int_{0}^{9} \frac{e ^{u} u}{9} d u

= \int_{0}^{9} \frac{e ^{u} u}{9} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int_{0}^{9} e^{u} u d u

Apply Integration By Parts

: [e^{u} u - \int e^{u} d u]_{0}^{9}

= \frac{1}{9} [e^{u} u - \int e^{u} d u]_{0}^{9}

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

= \frac{1}{9} [e^{u} u - e^{u}]_{0}^{9}

Compute the boundaries:

8 e^{9} + 1

= \frac{1}{9} (8 e^{9} + 1)

Simplify

= \frac{8 e ^{9} + 1}{9}

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What is the integral from 0 to 3 of xe^{3x} ?
The integral from 0 to 3 of xe^{3x} is (8e^9+1)/9