What is the integral from 0 to 4 of xe^{4x} ?

The integral from 0 to 4 of xe^{4x} is (15e^{16}+1)/(16)

integral from 0 to 4 of xe^{4x}

\int_{0}^{4} x e^{4 x} d x

\frac{15 e ^{16} + 1}{16}

Decimal

8330728.67547 \dots

Solution steps

\int_{0}^{4} x e^{4 x} d x

Apply u-substitution

= \int_{0}^{16} \frac{e ^{u} u}{16} d u

Take the constant out:

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

= \frac{1}{16} \cdot \int_{0}^{16} e^{u} u d u

Apply Integration By Parts

= \frac{1}{16} [e^{u} u - \int e^{u} d u]_{0}^{16}

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

= \frac{1}{16} [e^{u} u - e^{u}]_{0}^{16}

Compute the boundaries:

15 e^{16} + 1

= \frac{1}{16} (15 e^{16} + 1)

Simplify

= \frac{15 e ^{16} + 1}{16}

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What is the integral from 0 to 4 of xe^{4x} ?
The integral from 0 to 4 of xe^{4x} is (15e^{16}+1)/(16)