What is the integral of x^2e^{-5x} ?

The integral of x^2e^{-5x} is -1/125 (25e^{-5x}x^2-2(-5e^{-5x}x-e^{-5x}))+C

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integral of x^2e^{-5x}

\int x^{2} e^{- 5 x} d x

- \frac{1}{125} (25 e^{- 5 x} x^{2} - 2 (- 5 e^{- 5 x} x - e^{- 5 x})) + C

Solution steps

\int x^{2} e^{- 5 x} d x

Apply u-substitution

: \int - \frac{e ^{u} u ^{2}}{125} d u

= \int - \frac{e ^{u} u ^{2}}{125} d u

Take the constant out:

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

= - \frac{1}{125} \cdot \int e^{u} u^{2} d u

Apply Integration By Parts

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

= - \frac{1}{125} (u^{2} e^{u} - \int 2 u e^{u} d u)

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

= - \frac{1}{125} (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Substitute back

u = - 5 x

= - \frac{1}{125} ((- 5 x)^{2} e^{- 5 x} - 2 (e^{- 5 x} (- 5 x) - e^{- 5 x}))

Simplify

- \frac{1}{125} ((- 5 x)^{2} e^{- 5 x} - 2 (e^{- 5 x} (- 5 x) - e^{- 5 x})) : - \frac{1}{125} (25 e^{- 5 x} x^{2} - 2 (- 5 e^{- 5 x} x - e^{- 5 x}))

= - \frac{1}{125} (25 e^{- 5 x} x^{2} - 2 (- 5 e^{- 5 x} x - e^{- 5 x}))

Add a constant to the solution

= - \frac{1}{125} (25 e^{- 5 x} x^{2} - 2 (- 5 e^{- 5 x} x - e^{- 5 x})) + C

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

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

\int (3 x^{5} - 6 x + e^{x} - \frac{4}{x}) d x

\frac{\partial}{\partial x} (x cos (y) - y cos (x))

g (x) = ln (x e^{- 2 x})

What is the integral of x^2e^{-5x} ?
The integral of x^2e^{-5x} is -1/125 (25e^{-5x}x^2-2(-5e^{-5x}x-e^{-5x}))+C