1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Primitive function - smart substitution

Tags:
  1. Jan 5, 2017 #1

    Rectifier

    User Avatar
    Gold Member

    The problem
    $$ \int \frac{x}{\sqrt{x^2+2x+10}} \ dx $$

    The attempt

    ## \int \frac{x}{\sqrt{x^2+2x+10}} \ dx = \int \frac{x}{\sqrt{(x+1)^2+9}} \ dx##

    Is there any smart substitution I can make here to make this a bit easier to solve?

     
  2. jcsd
  3. Jan 5, 2017 #2

    Mark44

    Staff: Mentor

    Looks like this trig substitution might work, with ##\tan(u) = \frac {x + 1} 3##
     
  4. Jan 5, 2017 #3

    Rectifier

    User Avatar
    Gold Member

    Oh, okay so I basically get ##\int 3 \tan u - 1 \ du## after that and that is much easier. Very elegant substitution indeed.
     
  5. Jan 5, 2017 #4

    fresh_42

    Staff: Mentor

    The list of formulas suggests (by looking at the result) to first do an integration by parts to get rid of the ##x## in the nominator, and then some logarithm with ##z^2:=x^2+2x+10##. (But I only took a brief look.)
     
  6. Jan 5, 2017 #5

    Rectifier

    User Avatar
    Gold Member

    If we try to look for alternative substitutions;

    That integral from my problem looks a lot like an standard-integral which I have in my book:

    ## \int \frac{1}{\sqrt{x^2+a}} = \ln | x + \sqrt{x^2+a} | ## but the only thing different is the x in the nominator so I guess I could do this by integrating by parts and removing the x as a derivative inside the integral but I will end up with a part with ##...-\int {\ln | x + \sqrt{x^2+a}}| \ dx## which is not much easier to solve
     
  7. Jan 5, 2017 #6

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Writing the numerator as ##x = \frac{1}{2} (2x +2) -1## we have
    $$ \int \frac{x}{\sqrt{(x+1)^2+9}} \, dx = \frac{1}{2} \int \frac{d(x+1)^2}{\sqrt{(x+1)^2+9}} - \int \frac{dx}{\sqrt{(x+1)^2+9}}$$
    The first one has the form ##(1/2) \int du/\sqrt{u+9}##, while the second one has the form ##\int du/\sqrt{u^2+9}##.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Primitive function - smart substitution
  1. Substituting functions (Replies: 2)

Loading...