What is the integral of x^2*5^x ?

The integral of x^2*5^x is (5^xx^2)/(ln(5))-2/(ln(5))((5^xx)/(ln(5))-(5^x)/(ln^2(5)))+C

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integral of x^2*5^x

\int x^{2} \cdot 5^{x} d x

\frac{5 ^{x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{x} x}{ln ( 5 )} - \frac{5 ^{x}}{ln ^{2} ( 5 )}) + C

Solution steps

\int x^{2} \cdot 5^{x} d x

Apply Integration By Parts

: \frac{5 ^{x} x ^{2}}{ln ( 5 )} - \int \frac{2 \cdot 5 ^{x} x}{ln ( 5 )} d x

= \frac{5 ^{x} x ^{2}}{ln ( 5 )} - \int \frac{2 \cdot 5 ^{x} x}{ln ( 5 )} d x

\int \frac{2 \cdot 5 ^{x} x}{ln ( 5 )} d x = \frac{2}{ln ( 5 )} (\frac{5 ^{x} x}{ln ( 5 )} - \frac{5 ^{x}}{ln ^{2} ( 5 )})

= \frac{5 ^{x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{x} x}{ln ( 5 )} - \frac{5 ^{x}}{ln ^{2} ( 5 )})

Add a constant to the solution

= \frac{5 ^{x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{x} x}{ln ( 5 )} - \frac{5 ^{x}}{ln ^{2} ( 5 )}) + C

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What is the integral of x^2*5^x ?
The integral of x^2*5^x is (5^xx^2)/(ln(5))-2/(ln(5))((5^xx)/(ln(5))-(5^x)/(ln^2(5)))+C