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!

Integrals - the Substitution Rule with sin^n(x)

  1. Sep 11, 2012 #1
    1. The problem statement, all variables and given/known data
    Given that n is a positive integer, prove ∫sin^n(x)dx=∫cos^n(x)dx from 0 -> pi/2


    2. Relevant equations
    Perhaps sin^2(x)+cos^2(x)=1? Not sure.


    3. The attempt at a solution
    I honestly don't even know where to start. Should I set u=sin(x) or cos(x)? Doesn't seem to get the right answer either way...
     
  2. jcsd
  3. Sep 11, 2012 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    You need to show that the area under the graph from 0 to 90deg is the same for sin and cosine to any power.

    Your first step is to understand the problem - try sketching the graphs for a few powers and shading the area in question to see what you are up against.

    There is a rule for integrating powers of sin and cos ... you can derive it from integration by parts. Knowing the rule, you can probably just do it algebraically.

    Or you can try a substitution in one like [itex]x=\frac{\pi}{2}-u[/itex] and exploit the parity of the functions.
     
  4. Sep 11, 2012 #3
    Sorry, to clarify, the hint says:
    Use a trigonometric identity and substitution. Do not solve the definite integrals

    Given this information, how would you recommend solving?
     
  5. Sep 12, 2012 #4

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    The co-function identity states that [itex]\displaystyle \cos(x) = \sin\left(\frac{\pi}{2}-x\right)\ .[/itex]
     
  6. Sep 12, 2012 #5
    Thanks! I got so far:

    ∫cos^n(x)dx from 0 to pi/2
    =∫sin^n(pi/2-x)dx from 0 to pi/2
    =∫sin^n(-x)dx from -pi/2 to 0
    =∫sin^n(x)dx from 0 to pi/2

    Does this look right to you? Thanks for the hint!!
     
  7. Sep 12, 2012 #6

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Yeah, that's the idea - it was the second hint post #2 :)

    You should comment each step in your actual answer, to explain what you are doing.
     
  8. Sep 12, 2012 #7

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Use the substitution, [itex]u=\frac{\pi}{2}-x[/itex] to show that [itex]\displaystyle \int_{x=0}^{x=\pi/2}\sin^n(\frac{\pi}{2}-x)\,dx = \int_{u=\pi/2}^{u=0}-\sin(u)\,du\ .[/itex]
     
  9. Sep 12, 2012 #8
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Integrals - the Substitution Rule with sin^n(x)
Loading...