1. ## Integration problem

Hi, dear all:

I'm wondering how to integrate (exp(-sin(x)^2))*sin(x) with respect to x? I believe it has something to do with error function, but just can't figure out a right form. Pleas help me! Thank you very much.

simonwei

2. I'll post up where I'm at with this. Perhaps you will see it before I do.

Let $\displaystyle t=\sin(x)$, $\displaystyle dt = \cos(x)dx$.

Integral becomes,

$\displaystyle \int \frac{e^{-t^2}t}{\cos(\arcsin(t))} dt = \int \frac{e^{-t^2}t}{\sqrt{1-t^2}} dt$.

I further simplified this by letting $\displaystyle y=t^2$, $\displaystyle dy = 2tdt$.

Then we get...

$\displaystyle \int \frac{e^{-t^2}t}{\sqrt{1-t^2}} dt = \frac{1}{2} \int \frac{e^{-y}}{\sqrt{1-y}} dy$.

However I can't advance past this yet! Perhaps you can.

3. $\displaystyle e^{-\sin^2(x)}\sin(x) = e^{-1 + \cos^2(x) }\sin(x)$

$\displaystyle = e^{-1} e^{\cos^2(x) }\sin(x)$

Sub. $\displaystyle \cos(x) = t ~~~ -\sin(x)~dx = dt$

It becomes $\displaystyle -e^{-1 }\int e^{t^2}~dt$

It is not the error function but it is close , seemingly no one can use a finite number of elementary functions to express this integral ...

4. Nice solution simplependulum.

This is what I further thought of at 4am!

Let $\displaystyle y=t^2 - 1$ => $\displaystyle t^2 = y+1$

$\displaystyle 2tdt = dy$

Integral becomes...

$\displaystyle \int \frac{e^{-t^2}t}{\sqrt{1-t^2}}dt = \frac{1}{2} \int \frac{e^{-y}e^{-1}}{\sqrt{-y}}$

$\displaystyle = \frac{-ie^{-1}}{2} \int e^{-y}y^{-\tfrac{1}{2}} dy$

$\displaystyle = \frac{-i e^{-1} \sqrt{\pi}}{2} erf \Big{(}\sqrt{y}\Big{)}$

$\displaystyle = \frac{-i e^{-1} \sqrt{\pi}}{2} erf \Big{(}\sqrt{t^2 - 1}\Big{)}$

$\displaystyle = \frac{-i e^{-1} \sqrt{\pi}}{2} erf \Big{(}\sqrt{\sin^2(x) - 1}\Big{)}$

$\displaystyle = \frac{-i e^{-1} \sqrt{\pi}}{2} erf \Big{(}\sqrt{1 - \cos^2(x) - 1}\Big{)}$

$\displaystyle = \frac{-i e^{-1} \sqrt{\pi}}{2} erf(i\cos(x))$

EDIT: Apparently that minus should not be there (or there should be a minus in front of the $\displaystyle i\cos(x)$) but I can't figure out what I've done wrong after skimming over it.