Question

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.

Answer 1

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.

Answer 2

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)?