SOLUTION: I am trying to integrate the following: dx/(sin(x)+tan(x)) using u substitution. I know you can replace dx with 2du/1+u^2 and sin(x) can be replaced by 2u/1+u^2. If possible, p

Question 29614: I am trying to integrate the following:
dx/(sin(x)+tan(x))
using u substitution.
I know you can replace dx with 2du/1+u^2 and sin(x) can be replaced by 2u/1+u^2. If possible, please give every detail- I do not understand it and have been trying to do it for a while now! Thanks for your time!
Answer by shafkat_alam(3) (Show Source): You can put this solution on YOUR website!
(1 / (sin x + tan x) ) dx

= ((1/tan x)/((sin x/tan x)+1)) dx
= (sin x / (cos x (cos x + 1))) dx
now let u = cos x
so, -du = sin x dx
therefore, after substitution, the expression is,
((-1) / (u(u+1))) du
so, now partial ftactions are done,
so its it now becomes,
((1/(u+1)) - (1/u)) du
integrating this we get,
ln|u+1| - ln|u| + C
= ln|cos x + 1| - ln|cos x| + C
hope it helps.
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