Finding The Definite Integral

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1. Mar 27, 2017

doktorwho

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::

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. Mar 27, 2017

Incand

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.

3. Mar 27, 2017

Dick

I think you probably meant something more like $x=\pi-t$.

4. Mar 27, 2017

nuuskur

The substitution $x = \pi - t$ does indeed do the trick for proving the proposition before solving the given integral.

5. Mar 27, 2017

doktorwho

Yeah i did it with the substitution you proposed but how did you arrive at it?