Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integration by Parts confusions

  1. Apr 10, 2014 #1
    Hi all ! I'm new here :)

    So I'm facing some confusions here regarding integration by parts. While surfing through the internet to study more about this topic, I've came across two formulas which are used in solving problems related to integration by parts.
    They are
    1. uv - ∫uv'dx
    2. uv - ∫vdu

    May I know what are the difference between them and how should I use them in solving problems?

    Thanks a lot.
  2. jcsd
  3. Apr 10, 2014 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper

    Welcome to PF;
    Neither of them are complete ... to make sense of them you need to include the LHS:
    Your first one should have been... $$\int vu' \;dx = uv - \int uv' \;dx$$... this is integration by parts.

    But ##u' = du/dx## so the LHS becomes:$$\int vu' \;dx = \int v\frac{du}{dx} \;dx = \int v\;du$$
    Combine with the first one and you get: $$ \int v\;du = uv - \int uv' \;dx \implies \int uv' \;dx = uv-\int v\;du$$... which is your second one.

    Putting the two versions in the same form - for comparison:$$\begin{align}\int uv' \;dx &= uv-\int vu'\;dx \\ &= uv-\int v\; du\end{align}$$ ... they are just two different ways of writing the same thing.
  4. Apr 10, 2014 #3
    Thanks for the correction. Basically, either one can be used in problem solving. But I'm still a little confused with how this formula works in a whole. Can you please provide me with a brief explanation on the overall process? For example :

    ∫sin[itex]^{2}[/itex]2x dx

    My doubts are :

    let u= sin[itex]^{2}[/itex]2x and v'=1dx

    What happens next with
    ( u'= ) and ( v= )

    well, for now.


    edit : I used this website to check answers for my questions.
    as well as this one
    Last edited: Apr 10, 2014
  5. Apr 10, 2014 #4


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    Your post is a little unclear at the end. Are you asking how to take the derivative of sin[itex]^{2}[/itex](2x)? Are you asking how to integrate 1dx?
  6. Apr 10, 2014 #5


    User Avatar
    Science Advisor

    You understand that the prime indicates differentiation, don't you? if [itex]u= sin^2(2x)[/itex] the [itex]du= 4sin(2x) cos(2x)[/itex] by the chain rule. And if v'= dx (more correct notation would be "dv= dx") the v= x. So trying to apply "integration by parts" here would give
    [tex]\int u dv= uv- \int vdu[/tex]
    [tex]\int sin^2(x)dx= (sin^2(2x))(x)- 4\int x sin(2x)cos(2x)dx[/tex]

    That last integral does not look easier than the last so I would think this integral is NOT a good candidate for "integration by parts"!
    (If you really are concerned about integrating this particular function, you would be better advised to use the trig identity [itex]sin^2(\theta)= (1/2)(1- cos(2\theta))[/itex].)
  7. Apr 10, 2014 #6

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper

    note ##(vu)'=vu'+uv'## (this is just the product rule for differentiation.)
    if you integrate both sides you will end up with the integration by parts formula.
    The actual method though, is the clever choice of u and v'.

    You have to pick the u and v' so that the final integral is easier to solve than the initial one.
    It takes practice.

    There are plenty of examples online. i.e.
  8. Apr 11, 2014 #7
    I would like to understand how to take the derivative of sin[itex]^{2}[/itex](2x) as what they substituted is 2sin(4x) while what I derived from it was 2cos2xsin2x using chain rule.
  9. Apr 11, 2014 #8
    Exactly, I know that the prime indicated differentiation. My answer to the differential of u is exactly the same as what you got. However, some of the available online integral calculator got 2sin(4x) instead. This has been bugging me for the past few days.

    If its not a good candidate for integration by parts, are there any suggestions for the given example? NOTE : The integral calculator http://symbolab.com/solver/definite-integral-calculator/\int sin^{2}(2x)dx utilized integration by parts.

    Thanks a bunch, at least I know that the derivative of sin^2 (2x) that I calculated is correct. :)
  10. Apr 11, 2014 #9


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    The identity sin(A+B)=sin A cos B+cos A sin B can be used to rewrite 2sin(4x).
  11. Apr 13, 2014 #10
    I see, thanks for the heads up ! Guess this thread solved all my confusions bout integration by parts. I really appreciate your help!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Integration by Parts confusions
  1. Integration by parts (Replies: 7)

  2. Integration By Parts (Replies: 2)