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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = - y^{2}

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

Simplify

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

Add a constant to the solution

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

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