Question

1- cos x8.1 +cos x

Answer 1

The given function is

1−cosx 1+cosx .

∫ 1−cosx 1+cosx dx (1)

Also, 1−cosx=2 sin 2 x 2 and 1+cosx=2 cos 2 x 2 (2)

From (1),

1−cosx 1+cosx = 2 sin 2 x 2 2 cos 2 x 2 = tan 2 x 2 = sec 2 x 2 −1 (3)

Hence, from (3), we get,

∫ 1−cosx 1+cosx dx = ∫ ( sec 2 x 2 −1 )dx = ∫ sec 2 x 2 dx − ∫ 1 dx =2tan x 2 −x+c

Thus, the integral of the function 1−cosx 1+cosx is 2tan x 2 −x+c.