integrate the following

Question

Sec x/sec x+tan x

Answer 1

Question: Integrate sec(x)/(sec(x)+tan(x) )

sec(x)/(sec(x)+tan(x) )=1/(1+sin(x) )

1/(1+sin(x) )=1/(1+2∙sin(x/2)∙cos(x/2) )
=sec^2(x/2)/(sec^2(x/2)+2∙tan(x/2) )
=sec^2(x/2)/(1+tan^2(x/2)+2∙tan(x/2) )
=sec^2(x/2)/(1+tan(x/2) )^2

Now it is a simple integration.

∫sec^2(x/2)/(1+tan(x/2) )^2  dx=(-2)/(1+tan(x/2) )

Answer: ∫sec(x)/(sec(x)+tan(x) ) dx = (-2)/(1+tan(x/2) )