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

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

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integral of y^3e^{-y}

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

- e^{- y} y^{3} - 3 (e^{- y} y^{2} - 2 (- e^{- y} y - e^{- y})) + C

Solution steps

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

Apply u-substitution

: \int e^{u} u^{3} d u

= \int e^{u} u^{3} d u

Apply Integration By Parts

: u^{3} e^{u} - \int 3 u^{2} e^{u} d u

= u^{3} e^{u} - \int 3 u^{2} e^{u} d u

\int 3 u^{2} e^{u} d u = 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

= u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

Substitute back

u = - y

= (- y)^{3} e^{- y} - 3 ((- y)^{2} e^{- y} - 2 (e^{- y} (- y) - e^{- y}))

Simplify

(- y)^{3} e^{- y} - 3 ((- y)^{2} e^{- y} - 2 (e^{- y} (- y) - e^{- y})) : - e^{- y} y^{3} - 3 (e^{- y} y^{2} - 2 (- e^{- y} y - e^{- y}))

= - e^{- y} y^{3} - 3 (e^{- y} y^{2} - 2 (- e^{- y} y - e^{- y}))

Add a constant to the solution

= - e^{- y} y^{3} - 3 (e^{- y} y^{2} - 2 (- e^{- y} y - e^{- y})) + C

\int - \frac{14}{15} \cdot (x - 30) + 14 d x

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

\int \frac{x ^{2} + 25}{x ^{2}} d x

x = y^{2}, x = 2 - y

f (x) = \frac{- 4 x}{x ^{2} + 1}, (- 1, 2)

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