What is the integral of (x^2)/(e^{5x)} ?

The integral of (x^2)/(e^{5x)} is -1/5 e^{-5x}x^2+2/125 (-5e^{-5x}x-e^{-5x})+C

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integral of (x^2)/(e^{5x)}

\int \frac{x ^{2}}{e ^{5 x}} d x

- \frac{1}{5} e^{- 5 x} x^{2} + \frac{2}{125} (- 5 e^{- 5 x} x - e^{- 5 x}) + C

Solution steps

\int \frac{x ^{2}}{e ^{5 x}} d x

Apply Integration By Parts

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

\int - \frac{2}{5} e^{- 5 x} x d x = - \frac{2}{125} (- 5 e^{- 5 x} x - e^{- 5 x})

= - \frac{1}{5} e^{- 5 x} x^{2} - (- \frac{2}{125} (- 5 e^{- 5 x} x - e^{- 5 x}))

Simplify

= - \frac{1}{5} e^{- 5 x} x^{2} + \frac{2}{125} (- 5 e^{- 5 x} x - e^{- 5 x})

Add a constant to the solution

= - \frac{1}{5} e^{- 5 x} x^{2} + \frac{2}{125} (- 5 e^{- 5 x} x - e^{- 5 x}) + C

What is the integral of (x^2)/(e^{5x)} ?
The integral of (x^2)/(e^{5x)} is -1/5 e^{-5x}x^2+2/125 (-5e^{-5x}x-e^{-5x})+C