What is the integral of x^2e^{-14x} ?

The integral of x^2e^{-14x} is -1/1372 e^{-14x}(98x^2+14x+1)+C

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integral of x^2e^{-14x}

\int x^{2} e^{- 14 x} d x

- \frac{1}{1372} e^{- 14 x} (98 x^{2} + 14 x + 1) + C

Solution steps

\int x^{2} e^{- 14 x} d x

Apply u-substitution

: \int - \frac{e ^{u} u ^{2}}{2744} d u

= \int - \frac{e ^{u} u ^{2}}{2744} d u

Take the constant out:

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

= - \frac{1}{2744} \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}{2744} (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}{2744} (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Substitute back

u = - 14 x

= - \frac{1}{2744} ((- 14 x)^{2} e^{- 14 x} - 2 (e^{- 14 x} (- 14 x) - e^{- 14 x}))

Simplify

- \frac{1}{2744} ((- 14 x)^{2} e^{- 14 x} - 2 (e^{- 14 x} (- 14 x) - e^{- 14 x})) : - \frac{1}{1372} e^{- 14 x} (98 x^{2} + 14 x + 1)

= - \frac{1}{1372} e^{- 14 x} (98 x^{2} + 14 x + 1)

Add a constant to the solution

= - \frac{1}{1372} e^{- 14 x} (98 x^{2} + 14 x + 1) + C

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What is the integral of x^2e^{-14x} ?
The integral of x^2e^{-14x} is -1/1372 e^{-14x}(98x^2+14x+1)+C