Calculus III
posted by Sean on .
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.

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. 
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)?