What is the integral of x^2e^{-λx} ?

The integral of x^2e^{-λx} is -e^{-λx} 1/λ x^2+2/(λ^3)(-λe^{-λx}x-e^{-λx})+C

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integral of x^2e^{-λx}

\int x^{2} e^{- λ x} d x

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

Solution steps

\int x^{2} e^{- λ x} d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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\frac{d}{d x} (\frac{3 x ^{4} + 1}{x ^{3}})

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What is the integral of x^2e^{-λx} ?
The integral of x^2e^{-λx} is -e^{-λx} 1/λ x^2+2/(λ^3)(-λe^{-λx}x-e^{-λx})+C