Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

stottrupbailey

  • one year ago

integrate xcos^2(8x)dx

  • This Question is Closed
  1. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\int\limits_{}^{}xcos^2(8x)dx\]

  2. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I think I might need a half angle formula? I'm still confused though...

  3. LolWolf
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Yep, power-reducing formula, and then integration by parts. Use the fact that: \[ \cos^2(x)=\frac{1+\cos(2x)}{2} \]

  4. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    okay, so I've got it down to \[\int\limits_{}^{}x \frac{ 1 }{ 2 }(1+\cos(16x))dx\]

  5. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is that right?

  6. LolWolf
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    To make life easier, think of it in terms of powers of e: \[ e^{i\theta}=\cos(\theta)+i\sin(\theta) \]If you wish.

  7. LolWolf
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Yes, that is.

  8. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    can you give me a hint what to do next please?

  9. LolWolf
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Sure: try to integrate it using integration by parts. Do you know how to do this...?

  10. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yes I do. Should I use u=1+cos(16x) and dv=x ?

  11. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    dv=x dx

  12. LolWolf
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Try the other way around. Since we wish to reduce it to an integrable series. E.g. we can integrate \(\sin(l)\), but not \(\frac{x^2}{2}\sin(l)\), without integration by parts.

  13. LolWolf
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Try doing the following case: \[ dv=(1+\cos(16))dx,\;v=\cdots\\ u=x,\;du=dx \]

  14. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh, gotcha. okay just a sec

  15. LolWolf
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Quick tip, remember, intuitively, what integration by parts does: For some functions \(f, g\), integration by parts turns the problem: \[ \int f'\cdot g\;dx \]Into the problem: \[ \int f\cdot g'\;dx \]Think about this a little bit and it will become much easier to understand its use.

  16. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[x(x+\frac{ \sin(16x) }{ 16 })-\int\limits_{}^{}x+\frac{ \sin(16x) }{ 16 } dx\]

  17. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is that right so far?

  18. LolWolf
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Yessir/ma'am. Although don't forget the constant \(\frac{1}{2}\) that was at the beginning of the problem.

  19. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yep I left that hanging around somewhere. Okay, so without the 1/2 from before I got \[x^2+xsin(16x)/16 -\int\limits_{}^{}x+\frac{ \sin(16x) }{ 16 }dx\]

  20. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is that right?

  21. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    which would then be \[x^2+\frac{ xsin(16x) }{ 16 }-\frac{ x^2 }{ 2 }-\frac{ \cos(16x) }{ 32 }\]

  22. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oops that last one should be plus, not minus

  23. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ahhh nope that's not right just a sec

  24. LolWolf
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    You're really close, remember that the denominator has to multiply by 16, not add, and that the integral of sin is -cos, not +cos. But, you've got the jist. The devil is in the details.

  25. stottrupbailey
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Okay, I got the right answer. Thanks so much for the help!! :)

  26. LolWolf
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Sure thing, have a good one.

  27. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.