Integral of cos(e^x): Get Professional Advice

[rocomath]

\int\cos{e^x}dx

let

u=\cos{e^x}
du=-e^{x}\sin{e^x}dx

dv=dx
v=x

\int\cos{e^x}dx=x\cos{e^x}+\int xe^{x}\sin{e^x}dx

advice on any other approach?

 

[Hurkyl]

Well, that's generally what happens when you do the same integration by parts first forwards and then backwards: you wind up right where you started.

 

[Avodyne]

This does not have a closed-form solution in terms of simple functions. Are you sure you copied it correctly?

 

[rocomath]

This does not have a closed-form solution in terms of simple functions. Are you sure you copied it correctly?

yep, I'm in Calc 2, so i prob wouldn't be able to solve it even with a little guidance?

 

[Avodyne]

Well, there's not much to do. Make the substitution x=ln(u), dx=du/u, and you get cos(u)/u. The integral of this is the Cosine Integral function:
http://mathworld.wolfram.com/CosineIntegral.html

 

[SanjeevGupta]

What about expanding the cosine out and then integrating the infinite series. At least that gives you a result, however ugly.

 

[rocomath]

Well, there's not much to do. Make the substitution x=ln(u), dx=du/u, and you get cos(u)/u. The integral of this is the Cosine Integral function:
http://mathworld.wolfram.com/CosineIntegral.html

i was thinking of using this approach but i wasn't sure if i could do that

i also tried that website after class and i was like huh :D but thanks.

 

[rocomath]

What about expanding the cosine out and then integrating the infinite series. At least that gives you a result, however ugly.

i'm not on that section yet so i'll have to wait.

 

[real10]

ar u sure it's not cos(x)*e^x ?? for Calc 2 cos(e^x) seems kindof hard given that u haven't done series yet...