Integration Question (1 Viewer)

[Carrotsticks]

A student recently asked me a very interesting question:

Evaluate the integral:



I think the end result is really cool. I absolutely love problems like these.

The method I used is a bit unconventional (don't know how often it is used in the HSC), but perhaps some of you out there have alternative methods!

 

[Carrotsticks]

Hint:

 

[jyu]

2I=pi/2, I=pi/4

 

[RealiseNothing]

I'm guessing you use the fact that:



And from there either re-arrange the equation or somehow simplify.

I'll give it a proper go now though.

 

[Carrotsticks]

pi/4

Correct! A beautiful solution, do you think?

 

[jyu]

yes, I also like this type of questions

 

[RealiseNothing]

I got it, will post my solution. However since it's integrals, my latex may take a while.

 

[deswa1]

Quite interesting- thanks Carrot

 

[RealiseNothing]

=

Add them together:



Expand the bottom:



^^^Since

So the denominator = numerator. Hence the integral simply becomes:



Which is with limits



But this is double the answer since we added the two original integrals, so the actual answer is:



Awesome.

 

[Carrotsticks]

In case you are wondering why this is the case (the fact that if I sub n=1 or n=999999, I get the same answer), here is a diagram:

 

[bleakarcher]

Beautiful

 

[Nooblet94]

Very nice!