Posted by Sean on Thursday, March 13, 2008 at 2:43pm.
What is the indefinite integral of:
(12sin(3x)cscx)dx
If I could have two things given to me please, I'd be grateful. The two things are:
1) Trick/Technique to Use
2) First 2 or 3 steps
Thanks in advanced.

Calculus III  Count Iblis, Thursday, March 13, 2008 at 5:58pm
Express sin(3x) in terms of sin(x).
exp(ix) = cos(x) + i sin(x) (1)
Take the third power of both sides:
exp(3ix) = [cos(x) + i sin(x)]^3
From (1) it follows that the imaginary part of the left hand side is sin(3x)
Expand the right hand side and take the imaginary part. You then find:
sin(3x) = sin^3(x) + 3sin(x)cos^2(x)
= 3 sin(x)  4 sin^3(x)
It then follws that:
sin(3x)/sin(x) =
3  4 sin^2(x)
which is almost trivial to integrate.

Calculus III  Sean, Thursday, March 13, 2008 at 7:01pm
Out of curiosity, where did the 'i' come from, and what technique did you use?
Also, how did you write sin(3x) In terms of sin(x)?
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