What is the integral from 0 to 1 of x3^x ?

The integral from 0 to 1 of x3^x is (3ln(3)-2)/(ln^2(3))

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integral from 0 to 1 of x3^x

\int_{0}^{1} x 3^{x} d x

\frac{3 ln ( 3 ) - 2}{ln ^{2} ( 3 )}

Decimal

1.07364 \dots

Solution steps

\int_{0}^{1} x \cdot 3^{x} d x

Apply Integration By Parts

= [\frac{3 ^{x} x}{ln ( 3 )} - \int \frac{3 ^{x}}{ln ( 3 )} d x]_{0}^{1}

\int \frac{3 ^{x}}{ln ( 3 )} d x = \frac{3 ^{x}}{ln ^{2} ( 3 )}

= [\frac{3 ^{x} x}{ln ( 3 )} - \frac{3 ^{x}}{ln ^{2} ( 3 )}]_{0}^{1}

Compute the boundaries:

\frac{3 ln ( 3 ) - 2}{ln ^{2} ( 3 )}

= \frac{3 ln ( 3 ) - 2}{ln ^{2} ( 3 )}

What is the integral from 0 to 1 of x3^x ?
The integral from 0 to 1 of x3^x is (3ln(3)-2)/(ln^2(3))