What is the integral of y^2e^{-3y} ?

The integral of y^2e^{-3y} is -1/27 (9e^{-3y}y^2-2(-3e^{-3y}y-e^{-3y}))+C

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integral of y^2e^{-3y}

\int y^{2} e^{- 3 y} d y

- \frac{1}{27} (9 e^{- 3 y} y^{2} - 2 (- 3 e^{- 3 y} y - e^{- 3 y})) + C

Solution steps

\int y^{2} e^{- 3 y} d y

Apply u-substitution

: \int - \frac{e ^{u} u ^{2}}{27} d u

= \int - \frac{e ^{u} u ^{2}}{27} d u

Take the constant out:

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

= - \frac{1}{27} \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}{27} (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}{27} (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Substitute back

u = - 3 y

= - \frac{1}{27} ((- 3 y)^{2} e^{- 3 y} - 2 (e^{- 3 y} (- 3 y) - e^{- 3 y}))

Simplify

- \frac{1}{27} ((- 3 y)^{2} e^{- 3 y} - 2 (e^{- 3 y} (- 3 y) - e^{- 3 y})) : - \frac{1}{27} (9 e^{- 3 y} y^{2} - 2 (- 3 e^{- 3 y} y - e^{- 3 y}))

= - \frac{1}{27} (9 e^{- 3 y} y^{2} - 2 (- 3 e^{- 3 y} y - e^{- 3 y}))

Add a constant to the solution

= - \frac{1}{27} (9 e^{- 3 y} y^{2} - 2 (- 3 e^{- 3 y} y - e^{- 3 y})) + C

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What is the integral of y^2e^{-3y} ?
The integral of y^2e^{-3y} is -1/27 (9e^{-3y}y^2-2(-3e^{-3y}y-e^{-3y}))+C