What is the integral of x^23^x ?

The integral of x^23^x is (3^xx^2)/(ln(3))-2/(ln(3))((3^xx)/(ln(3))-(3^x)/(ln^2(3)))+C

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integral of x^23^x

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

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

Solution steps

\int 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

\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 )})

Add a constant to the solution

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

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