Troubleshooting Integral: Solving x * cos(x)dx with Ease

[dark_omen]

Hello,

I am having trouble with this integral, I don't know how to solve it.

integral(x * cos(x))dx
I tried it in the calculator and it gave me the integral back, and I don't know what method of integration to use to figure it out.
Well if anyone has a solution that would be great, thanks.

 

[berkeman]

What do you get when you differentiate x * sin(x)?

How about x^2 * sin(x)?

Do these give you some ideas about what you could differentiate to get to x * cos(x)?

 

[nrqed]

Hello,

I am having trouble with this integral, I don't know how to solve it.

integral(x * cos(x))dx
I tried it in the calculator and it gave me the integral back, and I don't know what method of integration to use to figure it out.
Well if anyone has a solution that would be great, thanks.

I would suggest integration by parts. There is on choice of "u" and "dv" that will lead to a very simple form after a single integration by part.s

 

[dark_omen]

Okay, I had to do integration by parts twice and I got:
((x^2 * cos(x))/6) + ((x^2 * cos(x))/3), I don't know if that is right though but it's what I came up with.

 

[Quaoar]

dark_omen, that answer is not correct.

Here's my solution:

u = x
dv = cos x dx

So:
du = dx
v = sin x

uv - I(v * du) = x sin x - I(sin x dx) = x sin x + cos x.

I've verified this answer in Mathematica as well.

 

[dark_omen]

Okay, so I guess I made the wrong choice in u and dv, and that's why my answer didn't come out right. Thanks.

 

[VietDao29]

dark_omen, that answer is not correct.

Here's my solution:

u = x
dv = cos x dx

So:
du = dx
v = sin x

uv - I(v * du) = x sin x - I(sin x dx) = x sin x + cos x.

I've verified this answer in Mathematica as well.

Why can't people just stop to give out COMPLETE solutions?!
?
Why?! What's so tempting about posting complete solutions like that?
Noone bothered to read https://www.physicsforums.com/showthread.php?t=28, eh?
And also, where has the Constant of Integration gone? Vanished?

 

[arildno]

Well, I find it excusable in that OP had shown quite a bit of work already.

 

[Quaoar]

Why can't people just stop to give out COMPLETE solutions?!
?
Why?! What's so tempting about posting complete solutions like that?
Noone bothered to read https://www.physicsforums.com/showthread.php?t=28, eh?
And also, where has the Constant of Integration gone? Vanished?

First you chide me for giving out the complete solution. Then you lecture me about not giving the complete solution... :) I thought that since he knew how to take an indefinite integral in the first place, he'd be smart enough to remember the +C.

Anyhow, no, I guess I missed that sentence in the rules. My apologies. Though I don't quite understand why it isn't left to the discretion of the person helping as to how much they want to help.