What is the integral of 6x^2e^{-x} ?

The integral of 6x^2e^{-x} is 6(-e^{-x}x^2+2(-e^{-x}x-e^{-x}))+C

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integral of 6x^2e^{-x}

\int 6 x^{2} e^{- x} d x

6 (- e^{- x} x^{2} + 2 (- e^{- x} x - e^{- x})) + C

Solution steps

\int 6 x^{2} e^{- x} d x

Take the constant out:

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

= 6 \cdot \int x^{2} e^{- x} d x

Apply Integration By Parts

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

= 6 (- e^{- x} x^{2} - \int - 2 e^{- x} x d x)

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

= 6 (- e^{- x} x^{2} - (- 2 (- e^{- x} x - e^{- x})))

Simplify

= 6 (- e^{- x} x^{2} + 2 (- e^{- x} x - e^{- x}))

Add a constant to the solution

= 6 (- e^{- x} x^{2} + 2 (- e^{- x} x - e^{- x})) + C

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What is the integral of 6x^2e^{-x} ?
The integral of 6x^2e^{-x} is 6(-e^{-x}x^2+2(-e^{-x}x-e^{-x}))+C