How Can I Integrate 1/(sin(x)+a) dx?

[msbell1]

Homework Statement

Integrate ∫dx/(sin(x)+a), where a is a constant.

Homework Equations

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
∫(sin(x)-a)/(sin²(x)-a²) dx.
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.

 

[ehild]

The integrand is a rational function of sin(x): rewrite it in terms of tan(x/2) and use the substitution tan(x/2)=t.
(sin(x)=2tan(x/2)/(1+tan²(x/2), cos(x)=(1-tan²(x/2)/(1+tan²(x/2). )

ehild