Question

`∫ dx/(cos x + sqrt3 sin x)` equals ______.

पर्याय

• log tan `(x/2 + π/12)` + C

• log tan `(x/2 − π/12)` + C

• `1/2 log tan (x/2 + π/12)` + C

• `1/2 log tan (x/2 − π/12)` + C

Answer 1

`∫ dx/(cos x + sqrt3 sin x)` equals `bbunderline(1/2 log tan (x/2 + π/12) + C)`.

Explanation:

I = `∫ (dx)/(cos x + sqrt3 sin x)`

⇒ I = `∫ (dx)/(2 [1/2 cos x + sqrt3/2 sin x])`

⇒ `1/2 ∫ dx/([sin  (π)/(6) cos x + cos  (π)/(6) sin x])`

⇒ `1/2.∫ dx/sin(x + π/6)`

⇒ I = `1/2.∫ "cosec"(x + π/6)dx`

∴ ∫ cosec x dx = log |(tan x/2)| + C

∴ I = `1/2.log tan (x/2 + π/12) + C`