What is the integral of x^2*5^{(x-1)} ?

The integral of x^2*5^{(x-1)} is 1/5 ((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-1)}

\int x^{2} \cdot 5^{(x - 1)} d x

\frac{1}{5} (\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 - 1} d x

Simplify

5^{x - 1} : 5^{x} \cdot 5^{- 1}

= \int x^{2} \cdot 5^{x} \cdot 5^{- 1} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5^{- 1} \cdot \int x^{2} \cdot 5^{x} d x

Apply exponent rule:

a^{- 1} = \frac{1}{a}

= \frac{1}{5} \cdot \int x^{2} \cdot 5^{x} d x

Apply Integration By Parts

= \frac{1}{5} (\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{1}{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{1}{5} (\frac{5 ^{x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{x} x}{ln ( 5 )} - \frac{5 ^{x}}{ln ^{2} ( 5 )})) + C

What is the integral of x^2*5^{(x-1)} ?
The integral of x^2*5^{(x-1)} is 1/5 ((5^xx^2)/(ln(5))-2/(ln(5))((5^xx)/(ln(5))-(5^x)/(ln^2(5))))+C