What is the integral from 0 to 2 of x*3^x ?

The integral from 0 to 2 of x*3^x is (18ln(3)-8)/(ln^2(3))

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integral from 0 to 2 of x*3^x

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

\frac{18 ln ( 3 ) - 8}{ln ^{2} ( 3 )}

Decimal Notation

9.75602 \dots

Solution steps

\int_{0}^{2} 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}^{2}

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

\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}^{2}

Compute the boundaries:

\frac{18 ln ( 3 ) - 8}{ln ^{2} ( 3 )}

= \frac{18 ln ( 3 ) - 8}{ln ^{2} ( 3 )}

\int cot^{3} (x) \cdot csc^{2} (x) d x

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\frac{d}{d x} (sin (x) + cos (x) + tan (x))

\frac{d}{d y} (ln (x y))

What is the integral from 0 to 2 of x*3^x ?
The integral from 0 to 2 of x*3^x is (18ln(3)-8)/(ln^2(3))