See attachment-- there may be a simpler way--I would definitely be interested.
That is the question. All that is there to it.
How do we proceed? I am thinking of sing double angle formla and making it sinx and sqrtcosx/2 in the denominator, bt not sre how that'll help.
Also, another thing on my mind is that if i substitute cos^2 (x/2) or Sin^2 (x/2) as another variable u, then du wold be in terms of sinxdx
First of all, thanks a lot!
I have two comments/queries/whatever....
What prompted you to go for that substitution? I know it is intuitive, but I am asking for a similarity somewhere in a more conventional integration problem which might have given you the hint.
The solution is pretty simple once the substitution is done, meaning the only bottleneck here is the substitution step, rest all being plane labour. Thanks again.
Say, if this is a simple solution, what methods would we be looking at for a "complex" solution? Just for the sake of curiosity?
To answer the first question --- you're right in saying that it is intuitive.
When you have a sqrt in the denominator if u = sqrt then du has
a 1/sqrt in it-- so I just thought if you use a u = cos^(-1/2) you get
a cos^(-3/2) when computing du. In all honesty I didn't know if it would
work when I first tried it.
As to the 2d question I have a feeling there may be a more clever
way of manipulating the integrand -- I should have said more clever not simpler