Help with evaluating indefinite integral

Use trig substitution. Set x = asin(theta) and sub in x and dx.

 

If you were to substitute [itex]x = a \sin(\theta)[/itex] then the limits x=-a to +a would change to theta = -pi/2 to pi/2. However I don't think that substitution will help all that much? It changes it into:
[tex]\int \frac{a^2 \cos^2 \theta}{b - \sin \theta}\, d\theta[/tex]
which doesn't look too easy.

 

That looks like a tricky integral pfsm. In the special case where b=a it simplifies (algebraically) to a very easy integral. For the general case however I'm stumped. Maybe if I show the simplified case it will give someone an else a lead.

[tex]\frac{\sqrt{a^2 - x^2}}{a-x} = \frac{\sqrt{a+x}}{\sqrt{a-x}} = \frac{a+x}{\sqrt{a^2-x^2}}[/tex]

So for this special case we have,
[tex] \int \frac{\sqrt{a^2 - x^2}}{a-x} \, dx = \int \frac{a}{\sqrt{a^2-x^2}} \, dx + \int \frac{x}{\sqrt{a^2-x^2}} \, dx[/tex]
which are both very easy integrals. For the general case however, hmmm, hopefully someone else can help out.

 

Ok so the question has now been changed to a definite integral, is that right? Sometimes there are more options available in how to handle a definite integral so it makes a difference (in general, though I'm not sure if it will make a difference this case, I'll take another look).