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

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

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integral from 1 to e of xe^{2x^2-1}

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

\frac{e ^{2 e^{2} - 1} - e}{4}

Decimal

240822.40084 \dots

Solution steps

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

Apply u-substitution

: \int_{1}^{2 e^{2} - 1} \frac{e ^{u}}{4} d u

= \int_{1}^{2 e^{2} - 1} \frac{e ^{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_{1}^{2 e^{2} - 1} e^{u} d u

Use the common integral:

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

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

Compute the boundaries:

e^{2 e^{2} - 1} - e

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

Simplify

= \frac{e ^{2 e^{2} - 1} - e}{4}

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