# Mathematics integrals question R(u,v) notation?

Hello. Can someone explain R(u,v) notation. Here is a link, page 12,

http://mathsforall.co.uk/userfiles/Integral%20Calc...

I don't understand the explanation at all.

how is uv-4 a rational function?

What does ∫R[sin(x),cos(x)]dx mean?

Why does ∫1/(tan(x) + sin(x))dx fall into this category?

Why does ∫[cos(x)]/[1 + cos(x)]dx fall into this category?

page 13, how is cos^2 x sin^3 x a rational function?

Does ∫R[sin(x),cos(x)]dx mean ∫1/(asin(x)+bcos(x)+c)dx?

Update:

Perhaps it would be easier if someone could exlplain just R(sinx) first then R(cosx) then R(sinx,cosx). How many sines can we have? How man cosines? of what degree? first degree only?

Relevance

R(sin x, cos x) essentially means that you have a ratio of polynomials in terms of sine and/or cosine.

For instance, 1/(tan(x) + sin(x)) = cos x/(sin x + sin x cos x) is a ratio of polynomials in terms of sine and cosine.

[More explicitly, the rational function is R(u, v) = v/(u + uv).

So, R(sin x, cos x) = cos x/(sin x + sin x cos x), as required.]

Similarly, [cos(x)]/[1 + cos(x)] is of the form R(u, v) = v/(1 + v) with u = sin x and v = cos x.

cos^2(x) sin^3(x) is rational in sin x and cos x, because polynomials are rational (with denominator 1):

R(u, v) = u^3 v^2 with u = sin x and v = cos x.

Finally, 1/(a sin(x)+b cos(x)+c) is of the form R(u, v),

with R(u, v) = 1/(au + bv + c) which is rational.

I hope this helps!