Are expressions such as sin(x^2) analytically integrable? I'm doing something for my calculus class (high school), and I'd like to use a trig function with a second degree parameter like the above (just one term, not a polynomial) but I can't figure out how to integrate it. My teacher says she thinks it is integrable, but doesn't remember how it would be done, and every table or textbook I've checked stays away from second degree composites inside trig functions. Does anyone know if this is possible? And if so, how is it done? Thanks! [edited by - Qoy on May 23, 2002 2:19:21 PM]

Matlab (symbolic toolbox) gives the value of as

1/2*2^(1/2)*pi^(1/2)*FresnelS(2^(1/2)/pi^(1/2)*x)

So as the previous poster said, you can use numerical integration on this function.

Best Regards,
- James

Thanks. The links and info were helpful, but I need to know if there's a method that does not require numerical methods. Unless I'm missing something here, all the methods suggested require numerical solutions, but I need something for the project that is purely analytical, otherwise I could just use a numerical integrator on the original function.

Or am I missing something?

[edited by - Qoy on May 23, 2002 12:37:23 AM]

Some functions simply don''t have explicit integrals or derivatives, or at least they haven''t been discovered yet. In those cases, numeric approaches are what you have to use.

Would a Taylor series suit your purposes? If you haven''t covered them, check your calc book. It might be in there.

Basically, sin(u) = u - (u^3)/3! + (u^5)/5! -(u^7)/7! + ...

Just have u = x^2 and integrate...

my math spidey sense says this might be possible.

you cant just look at something and say its not analytically solvable. or not analytically integratable. math people should be more rigorous then that. you have to prove the integration has no anayltic expression. other then that you say "i dont know how to integrate it" heck early newtonian physics had tons of functions they didnt know how to integrate that we teach how to integrate in high school now.

hrmmm

sin(x^2) = x^2 - x^6/3! + x^10/5! ...

after integration is:

x^3/3 - x^7/(7*3!) + x^11 (11*5!) ...

x (sin(x^2) f(x) hrmmm not quite =)

but now the real test is to see if we can find away to say the above can or cant have a finite expression. i took out finite polynomialexpression because i have hope that it could be expressed as the resultant of a trigonomic expansion. your gut should say thats possible because it was formed from a trigonomic expansion. thats not proof thats math spidey sense. manipulations on trigonomic expansions have a habit of coming back to trigonomic expansion via some profound geometery implication. but be darned if one comes to me.

edit: man i take forever to finish posting.






[edited by - declspec on May 24, 2002 1:24:30 AM]

Isn''t the integral of Sin(x^2)

(-cos(x^2))/(2x) ?

- cos(x^2)
-----------
2x