What is the integral from 1 to 6 of xe^{2x} ?

The integral from 1 to 6 of xe^{2x} is (11e^{12}-e^2)/4

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integral from 1 to 6 of xe^{2x}

\int_{1}^{6} x e^{2 x} d x

\frac{11 e ^{12} - e ^{2}}{4}

Decimal Notation

447573.82913 \dots

Solution steps

\int_{1}^{6} x e^{2 x} d x

Apply u-substitution

: \int_{2}^{12} \frac{e ^{u} u}{4} d u

= \int_{2}^{12} \frac{e ^{u} u}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int_{2}^{12} e^{u} u d u

Apply Integration By Parts

: [e^{u} u - \int e^{u} d u]_{2}^{12}

= \frac{1}{4} [e^{u} u - \int e^{u} d u]_{2}^{12}

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

= \frac{1}{4} [e^{u} u - e^{u}]_{2}^{12}

Compute the boundaries:

11 e^{12} - e^{2}

= \frac{1}{4} (11 e^{12} - e^{2})

Simplify

= \frac{11 e ^{12} - e ^{2}}{4}

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What is the integral from 1 to 6 of xe^{2x} ?
The integral from 1 to 6 of xe^{2x} is (11e^{12}-e^2)/4