Question

integration of sin2x/1+sin2x​

Answer 1

Answer:

Answer:

sin 2 x 1 + cos 2 x d x

ln  ( 1 + cos 2 x ) + C

Explanation:

At first glance, this one seems like a toughie - but it in fact can be solved with a clever application of trig identities.

Note that  

d d x cos 2 x = d d x ( cos x ) 2 = 2 ⋅ − sin x ⋅ cos x = − 2 sin  x  cos  x  

and  sin 2 x = 2 sin x cos x

These two results are almost exactly the same, differing only by a negative sign. But what does it mean in the course of our problem?

Well, look what happens when we let  

u = cos 2 x

-> We also need to replace  (d x)

In this  

u -substitution, as follows:

u = cos 2 x  d u d x = − 2 sin x cos x → d u = − 2 sin x cos x d x  

Before we apply this substitution, look at the modified integral (which has  

sin 2 x

-> Replaced with its equivalent  2 sin x cos x ): ∫ 2 sin x cos x 1 + cos 2 x d x