Just a few small Integration questions I have run into. (1 Viewer)

[Hazabot]

1. Is it true that -log(cosx) = log(secx)? As it says in this answer sheet that my Teacher gave me? If so why?

2. Are the proofs that abs(cos(theta)) = cos(theta) necessary in the HSC if so, can someone please explain them?

3. I noticed in the Standard Integrals sheet for Ext 2 2014 there was no cos^-1 integral, and I assumed this is because cos^-1 = - sin^-1 but apparently this isn't true (according to wolfram alpha) so Im not sure whether to just leave the answer as - sin^-1 or if there is a proof I need to do?

Cheers, everyone have a good holiday!

 

[InteGrand]

1. Is it true that -log(cosx) = log(secx)? As it says in this answer sheet that my Teacher gave me? If so why?
View attachment 32011
2. Are the proofs that abs(cos(theta)) = cos(theta) necessary in the HSC if so, can someone please explain them?
View attachment 32012
3. I noticed in the Standard Integrals sheet for Ext 2 2014 there was no cos^-1 integral, and I assumed this is because cos^-1 = - sin^-1 but apparently this isn't true (according to wolfram alpha) so Im not sure whether to just leave the answer as - sin^-1 or if there is a proof I need to do?

Cheers, everyone have a good holiday!

1. This is due to the log law . Let , and that identity falls out.

2. I think the HSC doesn't mind if you don't prove this, but it's better to prove it.

 

[Hazabot]

Thanks, I saw your post on my other thread, but I forgot to type a body :O