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

The integral of x^2e^{(x^2)/2} is e^{(x^2)/2}x-(pi/2)^{1/2}\erfi(x/(sqrt(2)))+C

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integral of x^2e^{(x^2)/2}

\int x^{2} e^{\frac{x ^{2}}{2}} d x

e^{\frac{x ^{2}}{2}} x - (\frac{π}{2})^{\frac{1}{2}} erfi (\frac{x}{2}) + C

Solution steps

\int x^{2} e^{\frac{x ^{2}}{2}} d x

Apply exponent rule:

\frac{a ^{n}}{b ^{n}} = (\frac{a}{b})^{n}

\frac{x ^{2}}{2} = (\frac{x}{2})^{2}

= \int x^{2} e^{(\frac{x}{2})^{2}} d x

Apply u-substitution

: \int 22 e^{u^{2}} u^{2} d u

= \int 22 e^{u^{2}} u^{2} d u

Take the constant out:

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

= 22 \cdot \int e^{u^{2}} u^{2} d u

Simplify

u^{2} : uu

= 22 \cdot \int e^{u^{2}} uu d u

Apply Integration By Parts

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

= 22 (\frac{1}{2} e^{u^{2}} u - \int \frac{1}{2} e^{u^{2}} d u)

\int \frac{1}{2} e^{u^{2}} d u = \frac{π}{4} erfi (u)

= 22 (\frac{1}{2} e^{u^{2}} u - \frac{π}{4} erfi (u))

Substitute back

u = \frac{x}{2}

= 22 (\frac{1}{2} e^{(\frac{x}{2})^{2}} \frac{x}{2} - \frac{π}{4} erfi (\frac{x}{2}))

Simplify

22 (\frac{1}{2} e^{(\frac{x}{2})^{2}} \frac{x}{2} - \frac{π}{4} erfi (\frac{x}{2})) : e^{\frac{x ^{2}}{2}} x - (\frac{π}{2})^{\frac{1}{2}} erfi (\frac{x}{2})

= e^{\frac{x ^{2}}{2}} x - (\frac{π}{2})^{\frac{1}{2}} erfi (\frac{x}{2})

Add a constant to the solution

= e^{\frac{x ^{2}}{2}} x - (\frac{π}{2})^{\frac{1}{2}} erfi (\frac{x}{2}) + C

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What is the integral of x^2e^{(x^2)/2} ?
The integral of x^2e^{(x^2)/2} is e^{(x^2)/2}x-(pi/2)^{1/2}\erfi(x/(sqrt(2)))+C