integration by parts: find int[-pi,pi][xsin(nx)]dx for all n

[lislr8]

Hi I have a question about integration by parts for all n in Natural numbers.

Using integration by parts, compute for each natural number n in N, Integral -pi to pi of xsin(nx) dx

I'll call the original integral K
I started off with dv=xdx, u=sin(nx), v= x^2/2, du= ncos(nx) dx
I used parts twice, and ended up with K = (some values we'll just call A) + (pi/n) K, so I rearranged to get K= nA/ n-pi provided that n does not equal pi.

Is that correct? The typing would be a lot more clear if I actually knew how to use this system. Sorry.

 

[Unco]

Re: integration by parts

Hi Lislr,

Your answer doesn't look quite right; remember that the value of cos(n*pi) depends on the parity of n. I suggest you switch your choices for u and dv around.

An acronym to (loosely) go by to remember which to choose for u is "I.L.A.T.E" - in order of the most preferred choice for u: Inverse trig functions, Logs, Algebraic expressions (like x, x^2), Trig functions, then Exponentials. In any case, if you have the possibility of choosing u=x, since u'=1 is quite nice that should be your first choice.

Show us your complete work if you get stuck.

 

[stapel]

The typing would be a lot more clear if I actually knew how to use this system. Sorry.

To learn how to use LaTeX, read the appropriate articles in the links in the "Forum Help" pull-down menu at the very top of every forum page. Alternatively, review on-topic articles in the "Administration Issues" category.

Thank you!

Eliz.

 

[galactus]

Try letting: \(\displaystyle u=x, \;\ dv=sin(nx)dx, \;\ du=dx, \;\ v=\frac{-cos(nx)}{n}\)

Now, it should fall into place.

 

[lislr8]

Okay I switched it but how do you evaluate it at -pi and pi? I end up getting something that I attached. So I was thinking that

if n=even, then it would be -2pi/n
if n=odd, then 2pi/n
and n can't be zero

So I downloaded the text edit but it's not letting me copy and paste into the box so I attached it. Thanks again.