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Question: \(\int_{}^{}\frac{dx}{\tan x + \cot x + \sec x + \cos ecx}\) =...

∫dxtan⁡x+cot⁡x+sec⁡x+cos⁡ecx\int_{}^{}\frac{dx}{\tan x + \cot x + \sec x + \cos ecx} =

A

12\frac{1}{2} (sin x – cos x) + c

B

12\frac{1}{2} (sin x – cos x – x) + c

C

12\frac{1}{2} (sin x – cos x + x) + c

D

None of these

Answer

12\frac{1}{2} (sin x – cos x – x) + c

Explanation

Solution

I = ∫sin⁡xcos⁡xdx1+sin⁡x+cos⁡x\int \frac { \sin x \cos x d x } { 1 + \sin x + \cos x } ̃ ∫sin⁡xsec⁡x+tan⁡x+1\int_{}^{}\frac{\sin x}{\sec x + \tan x + 1}dx

I = ∫sin⁡x(1+tan⁡x−sec⁡x)(1+tan⁡x)2−sec⁡2x\int \frac { \sin x ( 1 + \tan x - \sec x ) } { ( 1 + \tan x ) ^ { 2 } - \sec ^ { 2 } x } dx

̃

I = 12\frac{1}{2} ∫cos⁡x(1+tan⁡x−sec⁡x)\int_{}^{}{\cos x(1 + \tan x - \sec x)}dx

̃ 12\frac{1}{2} ∫\int(cosx + sin x – 1) dx

= 12\frac{1}{2} [sin x – cos x – x] + c