If the integral ∫ (5 tan x / tan x − 2)dx = x + a ln |sin x - 2 cos x| + k

• Integration

If the integral ∫ (5 tan x / tan x − 2)dx = x + a ln |sin x - 2 cos x| + k, then a is equal to

1. -1
2. 2
3. -2
4. 1

Answer

∫ (5 tan x /  tan x - 2)dx = ∫ (5 sin x / sin x - 2 cos x)dx

∫ [2(cos x + 2 sin x) + (sin x - 2 cos x)] / (sin x - 2 cos x)dx

2∫ [(cos x + 2 sin x) / (sin x - 2 cos x)]dx + ∫dx + k

 = 2 log |sin x – 2 cos x| + x + k  

∴ a = 2

The correct option is B.