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

The integral of e^{(-x)/a}x^2 is -ae^{-x/a}x^2+2a(-ae^{-x/a}x-a^2e^{-x/a})+C

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integral of e^{(-x)/a}x^2

\int e^{\frac{- x}{a}} x^{2} d x

- a e^{- \frac{x}{a}} x^{2} + 2 a (- a e^{- \frac{x}{a}} x - a^{2} e^{- \frac{x}{a}}) + C

Solution steps

\int e^{\frac{- x}{a}} x^{2} d x

Simplify

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

Apply Integration By Parts

: - a e^{- \frac{x}{a}} x^{2} - \int - 2 a e^{- \frac{x}{a}} x d x

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

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

= - a e^{- \frac{x}{a}} x^{2} - (- 2 a (- a e^{- \frac{x}{a}} x - a^{2} e^{- \frac{x}{a}}))

Simplify

= - a e^{- \frac{x}{a}} x^{2} + 2 a (- a e^{- \frac{x}{a}} x - a^{2} e^{- \frac{x}{a}})

Add a constant to the solution

= - a e^{- \frac{x}{a}} x^{2} + 2 a (- a e^{- \frac{x}{a}} x - a^{2} e^{- \frac{x}{a}}) + C

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What is the integral of e^{(-x)/a}x^2 ?
The integral of e^{(-x)/a}x^2 is -ae^{-x/a}x^2+2a(-ae^{-x/a}x-a^2e^{-x/a})+C