Solve Integration by Parts Homework: Find Error

[MitsuShai]

Homework Statement [/b]

http://i324.photobucket.com/albums/k327/ProtoGirlEXE/Capture1.jpg


The attempt at a solution[/b]
http://i324.photobucket.com/albums/k327/ProtoGirlEXE/100_0635.jpg
Answer I got: http://i324.photobucket.com/albums/k327/ProtoGirlEXE/Capture2.jpg

I thought my answer would be correct, but it was wrong. Where did I mess up at?

 

[Bohrok]

It looks good until you get to ¹/₂∫cos2x(2x)dx
You can't just integrate cos2x and 2x like that because they're multiplied together; you need to use integration by parts again since you have a product there. But before you do that, you can simplify it a bit by pulling the 2 from the (2x) out of the integrand and it will cancel with the 1/2 out front, giving you [STRIKE]∫xcosx dx[/STRIKE] ∫xcos2x dx. Then use u = x, [STRIKE]du = cosx dx[/STRIKE] dv = cos2x dx

 

[MitsuShai]

It looks good until you get to ¹/₂∫cos2x(2x)dx
You can't just integrate cos2x and 2x like that because they're multiplied together; you need to use integration by parts again since you have a product there. But before you do that, you can simplify it a bit by pulling the 2 from the (2x) out of the integrand and it will cancel with the 1/2 out front, giving you ∫xcosx dx. Then use u = x, du = cosx dx

you mean ∫xcos(2x) dx, right?
u=x dv=cos(2x)
du=dx v=(1/2)sin(2x)
(x^2 +1)((-1/2) cos(2x)) + x((1/2)sin(2x)) - ∫(1/2)sin(2x) dx
(x^2 +1)((-1/2) cos(2x)) + x((1/2)sin(2x)) - (1/2) ∫sin(2x) dx
(x^2 +1)((-1/2) cos(2x)) + x((1/2)sin(2x)) + (1/4) cos(2x) +c

Is that right? did I make an error?

 

[Bohrok]

Sorry about the errors, fixed now.
And yes, you got it right.