A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • 4 years ago

prove the following

  • This Question is Closed
  1. anonymous
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\int\limits_{x}^{2x}(1/t)dt\] is constant on interval (0, infinity)

  2. Phantom
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    hope u dont give a closed interval bro ...j/k..its anitiderivative in ln t. and ln (2x) - ln( x) = ln 2 ==a constant (+ve or -ve ??? its -ve)

  3. anonymous
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    So I realized the answer, *ahem* proof to this about an hour after I asked. If anyone saw this and wanted to know what to do, here are two proofs. Proof 1 (using 1st Fundamental Thm of Calc):\[F(x) = \int\limits_{x}^{2x}(1/t)dt\]\[F'(x)= [\ln |t|]_{x}^{2x} = \ln |2x| - \ln |x| = \ln|2x/x| = \ln 2 = 0\] Proof 2 (using 2nd Fundamental Thm of Calc):\[F(x) = \int\limits_{x}^{2x}(1/t)dt = \int\limits_{x}^{a}(1/t)dt + \int\limits_{a}^{2x}(1/t)dt = -\int\limits_{a}^{x}(1/t)dt + \int\limits_{a}^{2x}(1/t)dt\]\[F'(x) = -(1/x) + (1/2x)(2) = -(1/x) + (1/x) = 0\] Derivatives represent slope, so here F'(x) has zero or no slope. Therefore, it is constant (over all the domain of all positive real numbers).

  4. Phantom
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    ln2 is not 0

  5. anonymous
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh, you're right. idk, someone else I knew had that first proof...

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

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

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.