Integration of exponential and trigonometric forms

[AceAcke]

Homework Statement

http://d.imagehost.org/view/0659/Capture

Link to wolfram alfa:http://www.wolframalpha.com/input/?i=integrate%28cos%28e^x%29*e^x

What i don't understand is why whey do it like this and why i can't integrate by parts in this case?

Thanks for any replies!

 

[Whitishcube]

show us your steps for integration by parts. that way we can see if you may have made a mistake somewhere.

 

[Mark44]

Homework Statement

http://d.imagehost.org/view/0659/Capture

Link to wolfram alfa:http://www.wolframalpha.com/input/?i=integrate%28cos%28e^x%29*e^x

What i don't understand is why whey do it like this and why i can't integrate by parts in this case?

They use ordinary substitution because it is the most obvious method to try. Instead of asking why you can't use integration by parts, the question really should be "Why would I want to use integration by parts if there is a much simpler method I can use?"

It's possible that integration by parts will work here, but there aren't that many possible choices for u and dv. The thing about integration by parts is you want to choose dv so that it's not too simple (which eliminates dv = dx), but is still possible to integrate.

 

[AceAcke]

show us your steps for integration by parts. that way we can see if you may have made a mistake somewhere.

http://d.imagehost.org/view/0589/Capture2

They use ordinary substitution because it is the most obvious method to try. Instead of asking why you can't use integration by parts, the question really should be "Why would I want to use integration by parts if there is a much simpler method I can use?"

It's possible that integration by parts will work here, but there aren't that many possible choices for u and dv. The thing about integration by parts is you want to choose dv so that it's not too simple (which eliminates dv = dx), but is still possible to integrate.

The reson i did like that is because integration by parts is the chain rule in reverse right? so if I'm going to derive cos(e^x)*e^x then i would use the chain rule, and therefore i used integration by parts here

 

[Bohrok]

http://d.imagehost.org/view/0589/Capture2



The reson i did like that is because integration by parts is the chain rule in reverse right? so if I'm going to derive cos(e^x)*e^x then i would use the chain rule, and therefore i used integration by parts here

Integration by substitution is actually the chain rule in reverse. Integration by parts comes from the product rule of differentiation. So u-substitution is the way to go for this problem, although integration by parts might work.

 

[AceAcke]

ops that my bad, but still Cos(e^x)*e^x is in the form g(x)*f(x) which implies that integration by parts are in order?

 

[Whitishcube]

Well it should work, but cos(e^x) doesn't have an elementary antiderivative (I'm pretty sure), so when you tried to use integration by parts, you chose a dv which doesn't have an elementary antiderivative.

 

[AceAcke]

Okay, i have found the solution on wikipedia thanks to your replies.

The solution is to use substitution before using the integration by part method.

here is the link to wiki if anyone would come across the same problem.
http://en.wikipedia.org/wiki/Integration_by_parts#Integrals_with_powers_of_x_or_ex"

 

[Mark44]

Okay, i have found the solution on wikipedia thanks to your replies.

The solution is to use substitution before using the integration by part method.

I'm not sure exactly what you mean here. If you are saying that in this problem you should use substitution and then integration by parts, that is incorrect. The problem can be done using substitution alone. Integration by parts is completely unnecessary in this problem.

If you are saying that substitution is the preferred method here, then I agree.

here is the link to wiki if anyone would come across the same problem.
http://en.wikipedia.org/wiki/Integration_by_parts#Integrals_with_powers_of_x_or_ex"

 

[AceAcke]

I was to hasty in my previous post. no integration by parts is necessary!

http://d.imagehost.org/view/0054/Capture4

I think i need to go back to the basics to get my facts straight!

It funny how this expression cos(e^x)*e^x is easier to integrate than cos(e^x).

 

[Liquidxlax]

show us your steps for integration by parts. that way we can see if you may have made a mistake somewhere.

using parts is way over complicating things.Use substitution. u=e^x du=e^xdx

whoops should have read the whole thread