What is the integral of e^{-ax}x^2 ?

The integral of e^{-ax}x^2 is -e^{-ax} 1/a x^2+2/(a^3)(-ae^{-ax}x-e^{-ax})+C

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integral of e^{-ax}x^2

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

- e^{- a x} \frac{1}{a} x^{2} + \frac{2}{a ^{3}} (- a e^{- a x} x - e^{- a x}) + C

Solution steps

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

Apply Integration By Parts

: - e^{- a x} \frac{1}{a} x^{2} - \int - e^{- a x} \frac{2}{a} x d x

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

\int - e^{- a x} \frac{2}{a} x d x = - \frac{2}{a ^{3}} (- a e^{- a x} x - e^{- a x})

= - e^{- a x} \frac{1}{a} x^{2} - (- \frac{2}{a ^{3}} (- a e^{- a x} x - e^{- a x}))

Simplify

= - e^{- a x} \frac{1}{a} x^{2} + \frac{2}{a ^{3}} (- a e^{- a x} x - e^{- a x})

Add a constant to the solution

= - e^{- a x} \frac{1}{a} x^{2} + \frac{2}{a ^{3}} (- a e^{- a x} x - e^{- a x}) + C

f (x) = 3 - e^{- 3 x}, at x = - ln (2)

\frac{\partial}{\partial x} (20 + 5 sin (x) cos (x - y))

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

\frac{d}{d x} (\frac{3}{x ^{2} - 1})

What is the integral of e^{-ax}x^2 ?
The integral of e^{-ax}x^2 is -e^{-ax} 1/a x^2+2/(a^3)(-ae^{-ax}x-e^{-ax})+C