What is the integral of x^23^{2x} ?

The integral of x^23^{2x} is (9^xx^2)/(2ln(3))-1/(ln(3))((9^xx)/(2ln(3))-(9^x)/(4ln^2(3)))+C

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integral of x^23^{2x}

\int x^{2} 3^{2 x} d x

\frac{9 ^{x} x ^{2}}{2 ln ( 3 )} - \frac{1}{ln ( 3 )} (\frac{9 ^{x} x}{2 ln ( 3 )} - \frac{9 ^{x}}{4 ln ^{2} ( 3 )}) + C

Solution steps

\int x^{2} \cdot 3^{2 x} d x

Apply Integration By Parts

: \frac{9 ^{x} x ^{2}}{2 ln ( 3 )} - \int \frac{9 ^{x} x}{ln ( 3 )} d x

= \frac{9 ^{x} x ^{2}}{2 ln ( 3 )} - \int \frac{9 ^{x} x}{ln ( 3 )} d x

\int \frac{9 ^{x} x}{ln ( 3 )} d x = \frac{1}{ln ( 3 )} (\frac{9 ^{x} x}{2 ln ( 3 )} - \frac{9 ^{x}}{4 ln ^{2} ( 3 )})

= \frac{9 ^{x} x ^{2}}{2 ln ( 3 )} - \frac{1}{ln ( 3 )} (\frac{9 ^{x} x}{2 ln ( 3 )} - \frac{9 ^{x}}{4 ln ^{2} ( 3 )})

Add a constant to the solution

= \frac{9 ^{x} x ^{2}}{2 ln ( 3 )} - \frac{1}{ln ( 3 )} (\frac{9 ^{x} x}{2 ln ( 3 )} - \frac{9 ^{x}}{4 ln ^{2} ( 3 )}) + C

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What is the integral of x^23^{2x} ?
The integral of x^23^{2x} is (9^xx^2)/(2ln(3))-1/(ln(3))((9^xx)/(2ln(3))-(9^x)/(4ln^2(3)))+C