Proving Integration: Solving the Cosine and Sine Proofs for High School Students

[synkk]

I'm almost out of high school and have been trying to do some "harder" proofs, anyone, I'm not quite sure on how to proceed on this:

prove that for all k:

\displaystyle \int_0^{2\pi} (cos^{2k}x - sin^{2k} x) \ dx = 0

If anyone could start me off as I'll I have in my head is "try to express it into something you can integrate", but having no luck.

 

[SteamKing]

It's not clear what k is limited to. Is k all integers, all reals, etc? If k is an integer, will its range include negative values? What about k = 0?

 

[synkk]

It's not clear what k is limited to. Is k all integers, all reals, etc? If k is an integer, will its range include negative values? What about k = 0?

The parts before this question asked me to integrate cos^6(x) and sin^6x using identities from cos^6(x) - sin^6(x) and cos^6x + sin^6x which I done successfully. At the end of the question it says: "You might like to consider how you would prove that \displaystyle \int_0^{2\pi} (cos^{2k}x - sin^{2k} x) \ dx = 0 for all k, without having to derive a new identity for each value of k.

 

[tiny-tim]

hint: use symmetry

 

[synkk]

hint: use symmetry

Could you expand on that? I tried rewriting cos in terms of sin, but still pretty stuck

thanks

 

[tiny-tim]

cos^2k(π/2 - x) - sin^2k(π/2 - x) = … ?

 

[synkk]

cos^2k(π/2 - x) - sin^2k(π/2 - x) = … ?

= sin^{2k}x - cos^{2k}x = -(cos^{2k}x - sin^{2k} x)

I feel so dumb for not getting this :(

I tried for around 15 minutes playing around with what you gave me to show it's 0 but to no avail... I'll try again tomorrow and post if I've made progress. Thanks for your patience.

 

[Curious3141]

EDIT: Ignore my previous post, I'd made an error.

 

[Ray Vickson]

Could you expand on that? I tried rewriting cos in terms of sin, but still pretty stuck

thanks

Look at the graphs of sin(x) and cos(x). Can you get one of the graphs by shifting the other one to the right or the left?

 

[synkk]

π

Look at the graphs of sin(x) and cos(x). Can you get one of the graphs by shifting the other one to the right or the left?

yes cos(x) = sin(x + π/2)
also sin(x) = cos(x - π/2)

I can see why the integral is 0, by drawing both of the graphs and shading in the required area, but I'm having real trouble formalising a proof to actually put onto paper :\

 

[tiny-tim]

I can see why the integral is 0, by drawing both of the graphs and shading in the required area, but I'm having real trouble formalising a proof to actually put onto paper :\

try substituting y = π - x, or y = π/2 - x (and dy = -dx), and seeing what it does to the integral of one of your shaded parts