Integral of sin(theta)/(cos^2(theta)-2cos(theta))d(theta)

Question

Answer is ln(|cos(theta)/(sqrt(cos^2(theta)-2cos(theta)))|)

but I need the steps to get to the answer because I do not understand.

Answer 1

Make the substitution u = cos θ

Then the integrand becomes du/(2u-u²)

Then partial fractions (du/2)[(1/u)+(1/(2-u)]

This integrates as (1/2) ln u - (1/2)ln(2-u)

You can do the back substitution...but I do not agree with the answer as you have written it

This problem bothered me and I have done some more work on it. There are actually 2 solutions:

F(x)=.5ln[(cos x)/(cos x-2)] and G(x)=.5ln[(cos x)/(2-cos x)].

F'(x)=G'(x)=the given function to be integrated BUT F and G do NOT differ by a constant.

For odd multiples of π/2, both F and G are zero.

Between successive odd multiples of π/2 one and only one of F or G is defined.

Therefore the integral is only defined between successive odd multiples of π/2 and whether F or G is the

correct function to be used depends on the sign of the integrand. I know this is a long time after your queston was submitted but I do hope it helps our understanding.