What is the integral of x*5^x ?

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

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integral of x*5^x

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

Add a constant to the solution

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

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