Integrate ∫dx/(sin(x)+a), where a is a constant.
The Attempt at a Solution
I have been working on this for a while, and for some reason I can't figure it out. The attempt that seemed the most promising to me was to multiply top and bottom by (sin(x)-a), which gave
I could integrate the first term (∫sin(x)/(sin2(x)-a2)dx) by substituting 1-cos2(x) for sin2x, and then using partial fractions. However, the second term (-∫a/(sin2(x)-a2)dx) is causing me some trouble. Actually, with the second term, I again used partial fractions, but then I end up with (sin(x) +/- a) in the denominator, which ends up looking about the same as what I started with. Is there a substitution that would make this problem simple? Thanks.