Quantcast

A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

RolyPoly

  • one year ago

<Integration> \[\int (1+y^2)^\frac{5}{2} dy\] How to start?

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

    $$\int(1+y^2)^{5/2}\,dy=\int\left(\sqrt{1+y^2}\right)^5dy$$Now recognize this looks ready to use a trig substitution on:$$y=\tan\theta\implies dy=\sec^2\theta\ d\theta\\\int\left(\sqrt{1+\tan^2\theta}\right)^5\,dy=\int\sec^2\theta\left(\sqrt{\sec^2\theta}\right)^5\,d\theta=\int\sec^7\theta\,d\theta$$

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

    But that doesn't look good :(

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

    It's the best you're going to get...

  4. oldrin.bataku
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @Loser66 expanding a fractional power requires the generalized binomial theorem and results in an infinite series

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

    That's probably the best, thanks! :(

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

    @RolyPoly there are standard techniques to use here. Break into the product of powers of \(\sec^2\theta\) and tear stuff away.

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

    I know. I just couldn't believe that I have to change the into into trigo again, since I just got this from an trigo integral (\(\int csc^7x dx\)) :( Thanks again!

  8. oldrin.bataku
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Huh? that's weird @RolyPoly

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

    * \(-\int csc^7 x dx\) \[-\int csc^7 x dx\]\[=- \int (1+cot^2x)^\frac{5}{2}csc^2xdx\]\[=\int (1+cot^2x)^\frac{5}{2}d(cot x)\]\[=\int (1+y^2)^\frac{5}{2}dy\]

  10. oldrin.bataku
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @Loser66 actually brought something interesting and useful!$$\int(1+y^2)^2\sqrt{1+y^2}\,dy=\int(1+2y^2+y^4)\sqrt{1+y^2}\,dy$$Distribute and try integrating by term.

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

    hyperbolic substitution ... let y = sinh(u) \[ \int \cosh^5(u) \cosh u du = \int \cosh^6 u du = \frac{1}{2^6}\int (e^u + e^{-u})^6 du\]

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

    • Attachments:

Ask your own question

Sign Up
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.