Question

help step by step?

integral of (2sec^2(x))/(1+tan^2(x))dx

Answer 1

I want to be shown step by step.. explaining..

Answer 2

start using the identity
\[1+\tan^2 x= \sec^2 x\]

Answer 3

ok wait.

Answer 4

you can also do it via substitution:

integral of (2sec^2(x))/(1+tan^2(x))dx = 2 (integral of (sec^2(x))/(1+tan^2(x))dx) because 2 is a constant

2 (integral of (sec^2(x))/(1+tan^2(x))dx) ; u=tan(x) du= sec(x)^2

then you're left with 2 (integral of 1/(1+u^2)dx) ; you may notice this is equal to 2arctan(u)

substitute tanx back in for u, and you recieve 2arctan(tan(x)) = 2x

The most general version of this is 2x+C which is your answer
(sorry if the math text is a bit hard to read (I'm new here and really lost lol)