The substitution suggested requires a lot more information. I cannot supply all of it here.
It starts with tan (x/2) = sqrt[(1-cos x)/(1+cos x)] and eventually yields
sin x = 2t/(1+t2) and cos x=(1-t2)/(1+t2).... so make the substitution and after some algebra
the integrand then becomes 2 dt/[3(1+t2)-10]
Then make the substitution p = 1+t2 and you will get an integral of the form dy/y..
If this isn't enough help, make a comment & I will try to help further.
Later:
Integrand: 2dt/(1+t2) * 1/[3-5(2t/(1+t2)) ...expand the denominator in the 2nd term
Then 1+t2 divides out to give the result I stated originally...oops I missed a t in the denominator
2dt/[3(1+t2)-10t].
Now the denominator factors as (3t-1)(t-3)...so you will need to integrate by partial fractions
which will be a nuisance, but fairly easy to accomplish.
Ella C.
Hi Paul! Thank you for responding. I just realized that I forgot to include another key substitution- that sinx = 2t/(1+t^2), which you got in your answer. But may you flesh it out a bit more, especially regarding what comes next after making the substitution p = 1+t^2? Thank you so much!11/25/20