Results 1 to 5 of 5

Math Help - help with a trig substution integral

  1. #1
    Newbie
    Joined
    Jun 2009
    Posts
    10

    help with a trig substitution integral

    I'd appreciate if someone can help me out with the following integral:

    \int x*\sqrt{81-9x^2} dx

    I know this can be done with the power rule, but I'm looking to do it using trig substitution. I have a triangle set up, where sin theta = 3x/9, but get stuck halfway through the problem. Can anyone help me figure this one out? Thanks!
    Last edited by psu1024; June 22nd 2009 at 05:32 AM. Reason: Fixed latex closing tag: [/math] not [\math]
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jan 2009
    Posts
    715
     \int \frac{x~dx}{\sqrt{81-9x^2}}

    Substitute  x = 3 \sin{\theta}

     dx = 3 \cos{\theta} d \theta

     \int \frac{ (3 \sin{\theta} ) ~ 3 \cos{\theta} ~d \theta }{ 9 \sqrt{1 - \sin^2 \theta} }

     = \int \frac{ \sin{\theta} ~ \cos{\theta} ~d \theta  }{ \cos{\theta} }

     = \int  \sin{\theta} ~ d{\theta}
     = - \cos{\theta} + C

     = - \sqrt{ 1 - (\frac{x}{3})^2} + C

     = - \frac{ \sqrt{ 9 -x^2}}{3} + C
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,572
    Thanks
    1412
    I'd be interested in seeing how it can be done with the "power rule"!

    If you are wondering how simplependulum saw so quickly that he needed x= 3 sin( \theta), try factoring that "81" in the square root out:
    \int \frac{x dx}{\sqrt{81- 9x^2}}= \int\frac{xdx}{\sqrt{81(1- \frac{x^2}{9}})} =\int\frac{xdx}{9\sqrt{1- (\frac{x}{3})^2}} and to use " cos(\theta)= \sqrt{1- sin^2(\theta)}", we must have sin(\theta)= \frac{x}{3} or x= 3 sin(\theta).
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Jun 2009
    Posts
    10
    Thanks for the help from above posters, I appreciate the explanations. However, the original problem I had trouble with was
     \int x\sqrt{81-9x^2} dx ,
    with x times the sqrt, not x divided by the sqrt, which is part of what was tricky for me. (I think it was the problem I had with the Latex syntax when I originally posted that might have caused the mix-up.) So I've drawn my triangle, and I'm pretty sure that x=3 sin(\theta) and dx=3 cos(\theta) d\theta. I think that in theory, I should be able to figure out the rest and replace everything back into the equation, but I'm just stuck and unable to work through to get an answer. Is there something I am doing wrong? If not, how should I work this to get an answer?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by psu1024 View Post
    Thanks for the help from above posters, I appreciate the explanations. However, the original problem I had trouble with was
     \int x\sqrt{81-9x^2} dx ,
    with x times the sqrt, not x divided by the sqrt, which is part of what was tricky for me. (I think it was the problem I had with the Latex syntax when I originally posted that might have caused the mix-up.) So I've drawn my triangle, and I'm pretty sure that x=3 sin(\theta) and dx=3 cos(\theta) d\theta. I think that in theory, I should be able to figure out the rest and replace everything back into the equation, but I'm just stuck and unable to work through to get an answer. Is there something I am doing wrong? If not, how should I work this to get an answer?
    Take out the factor of 9 from the square root. Now substitute u = 9 - x^2. (This works for both your question and for the incorrect interpretation of your question).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Trig Integral
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 1st 2010, 06:09 PM
  2. solving differential equation using substution
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: March 5th 2010, 09:17 AM
  3. Difficult Substution Problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 12th 2009, 09:26 AM
  4. trig integral
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 7th 2009, 09:12 PM
  5. Replies: 3
    Last Post: September 9th 2008, 02:13 AM

Search Tags


/mathhelpforum @mathhelpforum