What is the integral from 0 to 6 of 1/3 xe^x ?

The integral from 0 to 6 of 1/3 xe^x is (5e^6+1)/3

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integral from 0 to 6 of 1/3 xe^x

\int_{0}^{6} \frac{1}{3} x e^{x} d x

\frac{5 e ^{6} + 1}{3}

Decimal

672.71465 \dots

Solution steps

\int_{0}^{6} \frac{1}{3} x e^{x} d x

Take the constant out:

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

= \frac{1}{3} \cdot \int_{0}^{6} x e^{x} d x

Apply Integration By Parts

: [e^{x} x - \int e^{x} d x]_{0}^{6}

= \frac{1}{3} [e^{x} x - \int e^{x} d x]_{0}^{6}

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

= \frac{1}{3} [e^{x} x - e^{x}]_{0}^{6}

Compute the boundaries:

5 e^{6} + 1

= \frac{1}{3} (5 e^{6} + 1)

Simplify

= \frac{5 e ^{6} + 1}{3}

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What is the integral from 0 to 6 of 1/3 xe^x ?
The integral from 0 to 6 of 1/3 xe^x is (5e^6+1)/3