Integral(sin(sin(x))*e^(sin^2(x)))

[Kanobaker]

This integral is supposed to go from -pi to pi. I have nothing to show from my work because I can't even manage to make sense of it. I have already tried multiple calculators, yet I come up emptyhanded. How does one go about breaking this expression down?

Edit: Any input on the expression: Integral(sin(x)*e^(x^2)), would also be helpful. I might be able to work with it then.

 

[skeeter]

[MATH]f(x) = \sin(\sin{x}) \cdot e^{\sin^2{x}}[/MATH]
[MATH]f(-x) = \sin[\sin(-x)]\cdot e^{[\sin(-x)]^2} = \sin(-\sin{x}) \cdot e^{[-\sin{x}]^2} = -\sin(\sin{x}) \cdot e^{\sin^2{x}} = -f(x)[/MATH]
the function is odd ...

[MATH]\int_{-a}^{a} f(x) \, dx = 0[/MATH]

... 2nd one this week.

 

[Kanobaker]

I can not for the life of me belive I missed that. Thank you so much, this is some major headache relief.

 

[skeeter]

Anytime I see a definite integral of the form [MATH]\int_{-a}^a f(x) \, dx[/MATH], and f(x) is nasty, the first thing I check is if f(x) is an odd function.

 

[Steven G]

Anytime I see a definite integral of the form [MATH]\int_{-a}^a f(x) \, dx[/MATH], and f(x) is nasty, the first thing I check is if f(x) is an odd function.

I don't even waste my time looking for nastiness, I immediately check to see if that integrand is odd or not. I think about it like this, if the integrand is not very complicated I can easily see within 1-2 seconds if it is odd or not. It is time well spent in my opinion.

 

[skeeter]

I don't even waste my time looking for nastiness, I immediately check to see if that integrand is odd or not. I think about it like this, if the integrand is not very complicated I can easily see within 1-2 seconds if it is odd or not. It is time well spent in my opinion.

Jomo ...