What is the integral of x5^{-x} ?

The integral of x5^{-x} is -(5^{-x}x)/(ln(5))-1/(ln^2(5))*5^{-x}+C

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integral of x5^{-x}

\int x 5^{- x} d x

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

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

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

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

Add a constant to the solution

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

\int x (2 x + 5) d x

\frac{\partial}{\partial x} (8 e^{x y})

\frac{d y}{d t} = - y^{2}

5 x ln (3 x) - 5 x

\frac{df}{d x} x = x^{3}

What is the integral of x5^{-x} ?
The integral of x5^{-x} is -(5^{-x}x)/(ln(5))-1/(ln^2(5))*5^{-x}+C