What is the integral of (1+s^2+2s)(e^{(s^2)/2}) ?

The integral of (1+s^2+2s)(e^{(s^2)/2}) is e^{(s^2)/2}s+2e^{(s^2)/2}+C

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integral of (1+s^2+2s)(e^{(s^2)/2})

\int (1 + s^{2} + 2 s) (e^{\frac{s ^{2}}{2}}) d s

e^{\frac{s ^{2}}{2}} s + 2 e^{\frac{s ^{2}}{2}} + C

Solution steps

\int (1 + s^{2} + 2 s) e^{\frac{s ^{2}}{2}} d s

\frac{s ^{2}}{2} = \frac{1}{2} s^{2}

= \int (1 + s^{2} + 2 s) e^{\frac{1}{2} s^{2}} d s

Expand

(1 + s^{2} + 2 s) e^{\frac{1}{2} s^{2}} : e^{\frac{s ^{2}}{2}} + e^{\frac{s ^{2}}{2}} s^{2} + 2 e^{\frac{s ^{2}}{2}} s

= \int e^{\frac{s ^{2}}{2}} + e^{\frac{s ^{2}}{2}} s^{2} + 2 e^{\frac{s ^{2}}{2}} s d s

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int e^{\frac{s ^{2}}{2}} d s + \int e^{\frac{s ^{2}}{2}} s^{2} d s + \int 2 e^{\frac{s ^{2}}{2}} s d s

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

\int e^{\frac{s ^{2}}{2}} s^{2} d s = e^{\frac{s ^{2}}{2}} s - (\frac{π}{2})^{\frac{1}{2}} erfi (\frac{s}{2})

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

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

Simplify

= e^{\frac{s ^{2}}{2}} s + 2 e^{\frac{s ^{2}}{2}}

Add a constant to the solution

= e^{\frac{s ^{2}}{2}} s + 2 e^{\frac{s ^{2}}{2}} + C

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