1. Limited time only! Sign up for a free 30min personal 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!

Integration after implicit differentiation

  1. Jan 16, 2009 #1
    1. The problem statement, all variables and given/known data

    What is the integral of:
    [tex] y + x \frac{dy}{dx}[/tex]

    3. The attempt at a solution
    I know that it is xy, after implicit differentiation.
    However, I cannot get it without prior knowledge of implicit differentiation.
  2. jcsd
  3. Jan 16, 2009 #2


    User Avatar
    Science Advisor

    Note that you would write the integral of that function as
    [tex]\int (y+ x\frac{dy}{dx})dx= \int ydx+ xdy[/tex]
    That means you are looking for a function F(x,y) so that [itex]dF= F_x dx+ F_y dy= ydx+ xdy[/itex]. If [itex]F_x= y[/itex] then integrating with respect to x, while keeping y constant, F(x,y)= xy+ g(y) where, because we are treating y as a constant, the "constant of integration" may depend on y: g(y). From that, [itex]F_y= x+ g'(y)= x[/itex] which tells us that g'(y)= 0. That means that g really is a constant: g= C so F(x,y)= xy+ C for any constant C.
  4. Jan 16, 2009 #3

    Do you mean this?
    [tex]\int (y+ x\frac{dy}{dx})dx= \int ydx+ \int xdy[/tex]
  5. Jan 16, 2009 #4
    Isn't that what he wrote?
  6. Jan 16, 2009 #5
    I have had an idea that the effect of integral sign ends at the first dx or dy sign. Perhaps, this is not true.
  7. Jan 16, 2009 #6
    Thank you, Hallsofivy!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Integration after implicit differentiation