Integrating Trigonometric Functions with Evasive Substitutions

[Rory9]

I am staring at an integral of the form

$$\int \frac{sin(at)}{(1 + bsin^{2}(at))^{1/2}} dt$$

which I have generated for myself (in attempting to model the behaviour of a particle in an oscillating field). I can't see a sensible substitution to try, at present. I could hunt down a standard integral, perhaps, but I suspect something obvious is evading me...

Any hints? Also, any suggestions for brushing up on solving integrals of this sort? I'm a bit rusty :-)

Cheers!

 

[Hurkyl]

I can't see a sensible substitution to try, at present.

It's sin that makes this complicated, isn't it? That seems the obvious place to start.

 

[Mark44]

This might be useful: Change 1 + b*sin^2(at) into 1 + b*(1 - cos^2(at) = 1 + b - b*cos^2(at).

You could then use the substitution u = cos(at), du = -a*sin(at)dt.

Then your integral would be roughly du/(A - bu^2)^(1/2), and you might be able to find that in a table of integrals or, failing that, apply a trig substitution.

Anyway, that's the direction I would go as a start.

 

[Rory9]

This might be useful: Change 1 + b*sin^2(at) into 1 + b*(1 - cos^2(at) = 1 + b - b*cos^2(at).

You could then use the substitution u = cos(at), du = -a*sin(at)dt.

Then your integral would be roughly du/(A - bu^2)^(1/2), and you might be able to find that in a table of integrals or, failing that, apply a trig substitution.

Anyway, that's the direction I would go as a start.

A good suggestion. Thank you. It will end up with an arc sin of a cos, I think, but perhaps that can be rewritten more elegantly...