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Finding The Definite Integral

  1. Mar 27, 2017 #1
    1. The problem statement, all variables and given/known data
    Today i had a test on definite integrals which i failed. The test paper was given to us so we can practise at home and prepare better for the next one. This is the first problem which i need your help in solving::
    Test.JPG

    2. Relevant equations
    3. The attempt at a solution

    As no points were given for a solution of the below integrals without the proof of the integral property above i need to do that first. I had no idea how to start the proof. I figured i need to use some sort of substitution but i fail to see which and why. Could you give me a hint on how to do this? I know i haven't provided any work done by myself but i can't since i can't start. I didn't have a clue calculus was going to be this hard :/.
    Thanks
     
  2. jcsd
  3. Mar 27, 2017 #2
    The key here is that ##sin(x)## is symmetric around ##pi/2##. Hence the substitution ##t=pi/2+x## may be of use. You then see that a term in your new expression should disappear.
     
  4. Mar 27, 2017 #3

    Dick

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    Homework Helper

    I think you probably meant something more like ##x=\pi-t##.
     
  5. Mar 27, 2017 #4
    The substitution ##x = \pi - t ## does indeed do the trick for proving the proposition before solving the given integral.
     
  6. Mar 27, 2017 #5
    Yeah i did it with the substitution you proposed but how did you arrive at it?
     
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