What is the integral from 0 to 2 of x3^{-x^2} ?

The integral from 0 to 2 of x3^{-x^2} is (40)/(81ln(3))

integral from 0 to 2 of x3^{-x^2}

\int_{0}^{2} x 3^{- x^{2}} d x

\frac{40}{81 ln ( 3 )}

Decimal

0.44950 \dots

Solution steps

\int_{0}^{2} x \cdot 3^{- x^{2}} d x

Apply u-substitution

= \int_{1}^{\frac{1}{81}} - \frac{1}{2 ln ( 3 )} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{\frac{1}{81}}^{1} - \frac{1}{2 ln ( 3 )} d u

Integral of a constant:

\int a d x = a x

= - [(- \frac{1}{2 ln ( 3 )}) u]_{\frac{1}{81}}^{1}

Simplify

= - [- \frac{1}{2 ln ( 3 )} u]_{\frac{1}{81}}^{1}

Compute the boundaries:

- \frac{40}{81 ln ( 3 )}

= - (- \frac{40}{81 ln ( 3 )})

Simplify

= \frac{40}{81 ln ( 3 )}

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What is the integral from 0 to 2 of x3^{-x^2} ?
The integral from 0 to 2 of x3^{-x^2} is (40)/(81ln(3))