What is the integral of y^3e^{(y^2)/2} ?

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

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integral of y^3e^{(y^2)/2}

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

e^{\frac{y ^{2}}{2}} y^{2} - 2 e^{\frac{y ^{2}}{2}} + C

Solution steps

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

Apply u-substitution

: \int \frac{e ^{\frac{u ^{\frac{2}{3}}}{2}} u ^{\frac{1}{3}}}{3} d u

= \int \frac{e ^{\frac{u ^{\frac{2}{3}}}{2}} u ^{\frac{1}{3}}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int e^{\frac{u ^{\frac{2}{3}}}{2}} u^{\frac{1}{3}} d u

Apply u-substitution

: \int \frac{3 e ^{\frac{v}{2}} v}{2} d v

= \frac{1}{3} \cdot \int \frac{3 e ^{\frac{v}{2}} v}{2} d v

Take the constant out:

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

= \frac{1}{3} \cdot \frac{3}{2} \cdot \int e^{\frac{v}{2}} v d v

Apply u-substitution

: \int 4 e^{w} w d w

= \frac{1}{3} \cdot \frac{3}{2} \cdot \int 4 e^{w} w d w

Take the constant out:

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

= \frac{1}{3} \cdot \frac{3}{2} \cdot 4 \cdot \int e^{w} w d w

Apply Integration By Parts

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

= \frac{1}{3} \cdot \frac{3}{2} \cdot 4 (e^{w} w - \int e^{w} d w)

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

= \frac{1}{3} \cdot \frac{3}{2} \cdot 4 (e^{w} w - e^{w})

Substitute back

= \frac{1}{3} \cdot \frac{3}{2} \cdot 4 e^{\frac{( y ^{3} ) ^{\frac{2}{3}}}{2}} \frac{( y ^{3} ) ^{\frac{2}{3}}}{2} - e^{\frac{( y ^{3} ) ^{\frac{2}{3}}}{2}}

Simplify

\frac{1}{3} \cdot \frac{3}{2} \cdot 4 e^{\frac{( y ^{3} ) ^{\frac{2}{3}}}{2}} \frac{( y ^{3} ) ^{\frac{2}{3}}}{2} - e^{\frac{( y ^{3} ) ^{\frac{2}{3}}}{2}} : e^{\frac{y ^{2}}{2}} y^{2} - 2 e^{\frac{y ^{2}}{2}}

= e^{\frac{y ^{2}}{2}} y^{2} - 2 e^{\frac{y ^{2}}{2}}

Add a constant to the solution

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

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