What is the integral from 0 to 2 of x^23^x ?

The integral from 0 to 2 of x^23^x is (36ln^2(3)-36ln(3)+16)/(ln^3(3))

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integral from 0 to 2 of x^23^x

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

\frac{36 ln ^{2} ( 3 ) - 36 ln ( 3 ) + 16}{ln ^{3} ( 3 )}

Decimal

15.00798 \dots

Solution steps

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

Apply Integration By Parts

: [\frac{3 ^{x} x ^{2}}{ln ( 3 )} - \int \frac{2 \cdot 3 ^{x} x}{ln ( 3 )} d x]_{0}^{2}

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

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

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

Compute the boundaries:

\frac{36 ln ^{2} ( 3 ) - 36 ln ( 3 ) + 16}{ln ^{3} ( 3 )}

= \frac{36 ln ^{2} ( 3 ) - 36 ln ( 3 ) + 16}{ln ^{3} ( 3 )}

\frac{d}{d x} (\frac{7 cos ( x ) - 8}{sin ( x )})

\int 3 csc (x) cot (x) d x

\int \frac{x}{x ^{4} + 10} d x

\frac{x}{x - 1}

\frac{d}{d x} (x (1 - x)^{2})

What is the integral from 0 to 2 of x^23^x ?
The integral from 0 to 2 of x^23^x is (36ln^2(3)-36ln(3)+16)/(ln^3(3))