Definite integration by parts!

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  • #1
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Hello there. I feel like this isn't the right answer, but I'd like some verification as to where exactly I went wrong! 1. Homework Statement is [tex]\int_{0}^{pi}x^2cos x dx[/tex]

3. The Attempt at a Solution went something like this:
[tex]u=x^2 dv=cos x dx
du=2x dx v=\int_{0}^{pi}cos x dx= sin x[/tex]

The integral was then:
[tex]\int_{0)^{pi}x^2cos x dx= x^2 sin x\right]_{0}^{pi}-\int_{0}^{pi}sin x 2x dx[/tex]

to solve:

[tex]=x^2 sin x + cos x x^2[/tex]
[tex]=pi^2 sin pi + cos 0 o^2[/tex]
[tex]9.87 times 0=1+0[/tex]
[tex]=1[/tex]

Thanks for all your help in advance! :smile:





 
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Answers and Replies

  • #2
cristo
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Hello there. I feel like this isn't the right answer, but I'd like some verification as to where exactly I went wrong! 1. Homework Statement is [tex]\int_{0}^{pi}x^2cos x dx[/tex]

3. The Attempt at a Solution went something like this:
[tex]u=x^2 \hspace{5mm} dv=cos x dx \hspace{5mm}
du=2x dx \hspace{5mm} v=\int_{0}^{pi}cos x dx= sin x[/tex]

The integral was then:
[tex]\int_{0}^{pi}x^2cos x dx= \left[x^2 sin x\right]_{0}^{\pi}-\int_{0}^{\pi}sin x 2x dx[/tex]

to solve:

[tex]=x^2 sin x + cos x x^2[/tex]
[tex]=\pi^2 sin pi + cos 0 o^2[/tex]
[tex]9.87 \times 0=1+0[/tex]
[tex]=1[/tex]

Thanks for all your help in advance! :smile:





Ive just cleaned up your tex so I can check your work.
 
  • #3
cristo
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From here: [tex]\int_{0}^{\pi}x^2cos x dx= \left[x^2 sin x\right]_{0}^{\pi}-\int_{0}^{\pi}sin x 2x dx[/tex]

You need to apply integration by parts again to the second term on the RHS. I don't quite know what you've done in your original post.
 
  • #4
D H
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You started just fine:

[tex]\int x^2 \cos x\,dx = x^2\sin x - \int \sin x\,2x\,dx[/tex]

You're problem is the second integration:

[tex]\int \sin x\,2x\,dx \ne -\cos x\,x^2[/tex]
 
  • #5
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Do I need to integrate by parts again?
[tex]\int \sin x\,2x,dx=2/int,x,sinx,dx[/tex]
so
[tex]= \left[x^2 sin x\right]_{0}^{\pi}-2\int_{0}^{\pi}xsin x dx[/tex]
is this right?
 
  • #6
Gib Z
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No :(

[tex]\int \sin x 2x dx[/tex].
u=sin x, dv = 2x dx
du=cos x dx, v=x^2

Subbing these straight into integration by parts it should be
[tex]\sin (x) \cdot x^2 - \int x^2 \cos x dx[/tex].

Hmm Isnt that interesting, your original integral appeared again, how could this help i wonder :P
 
  • #7
Gib Z
Homework Helper
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Actually I did it wrong, you should be able to see why the way I did it doesn't help.

instead, in integration by parts, let u=2x and dv= sin x dx
that way, du= 2 dx, and v= -cos x, which gives us
[tex]-2x\cos x + 2\int \cos x dx[/tex].
The integral in that is just 2 sin x, so the solution to your original integral is easy from here.
 

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