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

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

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

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

= \frac{1}{a} (- 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}})

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

Simplify

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

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (e^{(t - x)})

\frac{d}{d x} (4 x^{2} - 5)

y = 3 x^{3} - 10 x - \frac{4}{x ^{3}}

\frac{\partial}{\partial x} (\frac{3 x y}{x ^{4} + y ^{4}})

\frac{\partial}{\partial x} (x + 2 y + 1)

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