Let , then
Now use the substitutions , , and . You'll find that some really good simplification happens.
I also have limits - zero and Pi.
I seem to have obtained an expression for the integral but substituting Pi makes it troublesome.
so, (apologies for the lack of LaTeX)
let u = tan(x/2)
u = sin(x/2) / cos(x/2)
u cos(x/2) = sin(x/2)
u^2 cos^2(x/2) = sin^2(x/2)
u^2 cos^2(x/2) = (1 - cos^2(x/2)
cos^2(x/2)(1 + u^2) = 1
cos^2(x/2) = 1/ (1 + u^2)
sin^2(x/2) = 1 - 1 /(1 + u^2) = u^2/(1 + u^2)
cos x = cos^2(x/2) - sin^2(x/2)
= 1 / (1 + u^2) - u^2/(1 + u^2) = (1 - u^2)/(1 + u^2)
cos x = (1 - u^2)/(1 + u^2)
when u = tan(x/2)
x/2 = arctan(u)
dx = 2du / (1 + u^2)
substituting these into integral
∫ dx /(5 + 4 cos x)
= ∫ 2du /(1 + u^2)(5 + 4(1 - u^2)/(1 + u^2)
= ∫ 2du/ (5 + 5u^2 + 4 - 4u^2)
= ∫ 2du/ (9 + u^2)
= 2/9 ∫du/ [1 + (u/3)^2)]
= (2/3) ∫(du/3)/ [1 + (u/3)^2)]
= (2/3) arctan(u/3) + C
substitute u/3 = (1/3)tan(x/2)
= (2/3) arctan[1/3tan(x/2) ] + C
but putting Pi into tan(x/2) makes the expression undefined! where did i go wrong?
It's too hard to read. Check out this link: Learn Latex so we don't go crazy looking at that stuff.
So your integrand Looks like this: