Question
Question: \(\int_{}^{}\sqrt{1 + \cos ecx}\)dx =...
∫1+cosecxdx =
A
± sin–1 (tan x – sec x) + c
B
2sin–1 (cos x) + c
C
sin–1(cos2x−sin2x) + c
D
± 2 sin–1(sin2x−cos2x) + c
Answer
± 2 sin–1(sin2x−cos2x) + c
Explanation
Solution
I = dx
I = dx = ± ∫2sin2xcos2xsin2x+cos2xdx
I = ± ∫1−(sin2x−cos2x)2sin2x+cos2xdx
Put sin 2x – cos 2x = t, (cos2x+sin2x) dx = 2dt
I = ± ∫1−t22dt = ± 2 sin–1(t)
= ± 2 sin–1 (sin2x−cos2x) + c