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How does one integrate u√(1+u^2) du?

  1. Jul 14, 2015 #1
    1. The problem statement, all variables and given/known data
    How does one integrate u√(1+u^2) du?


    2. Relevant equations
    Forget what u is being substituted for. It doesn't really matter right now. I just need to integrate what is typed above.

    3. The attempt at a solution
    I am completely lost and not sure what to do.
     
  2. jcsd
  3. Jul 14, 2015 #2

    SammyS

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    What's the derivative of 1+u2 ?
     
  4. Jul 14, 2015 #3
    The derivative of 1+u^2 with respect to u is 2u du.

    I know the answer is (1/3)(1 + u^2)^(3/2) but I don't know how to find it.
     
  5. Jul 14, 2015 #4

    SammyS

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    Do you know the method of substitution ?
     
  6. Jul 14, 2015 #5
    No.
     
  7. Jul 14, 2015 #6

    SammyS

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    That should have been a BIG hint.

    Take the derivative of (1/3)(1 + u^2)^(3/2) . Then work backwards.
     
  8. Jul 14, 2015 #7
    Thanks for the tip. I did that but I still fail to see the connection. Can you, or someone else, give me the next step?
     
  9. Jul 14, 2015 #8
    Let v= 1+u^2
    so dv=2u du

    how do you replace du and (1+u^2)
     
  10. Jul 14, 2015 #9
    Numerator? There is no fraction here unless you you consider 1 as the denominator.
     
  11. Jul 14, 2015 #10
    ahh, thought I saw a / in front of the root.
    fixed it. That partial square root symbol gets me
     
  12. Jul 15, 2015 #11
    That helped immensely. Thank you so much. I forgot that I could use substitution more than once.
     
  13. Jul 15, 2015 #12

    SammyS

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    So, you do know substitution.
     
  14. Jul 15, 2015 #13

    epenguin

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    Of course a substitution in a substitution is still one substitution overall so you can substitute twice all at once, but maybe it is better to go by stages at first. :oldsmile:
     
  15. Jul 15, 2015 #14
    Yes, I know substitution. You asked me if I knew the method of substitution, which I understood as, you asking if I knew which method of substitution that I should use.
     
  16. Jul 16, 2015 #15
    1 + [sinh^2(x)] = [cosh^2(x)]
     
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