Question

Evaluate \[ \int \frac{dx}{\sin\left(x-\frac{\pi}{3}\right)\cos x} \]

• A. \(2\log\left(\dfrac{\tan x-\sqrt{3}}{2}\right)+C\)
• B. \(2\log\left(\sin\left(x-\dfrac{\pi}{3}\right)\cos x\right)+C\)
• C. \(2\log\left(\tan x+\sqrt{3}\right)+C\)
• D. \(2\log\left(\sin x+\sqrt{3}\cos x\right)+C\)

Hint:

When an integral contains \(\sin(x-a)\cos x\), first expand \(\sin(x-a)\), then express the result in terms of \(\tan x\) and use \(d(\tan x)=\sec^2x\,dx\).

Answer 1

The Correct Option is A

Step 1: Simplify the trigonometric expression.
We need to evaluate \[ I=\int \frac{dx}{\sin\left(x-\frac{\pi}{3}\right)\cos x} \] Using the identity \[ \sin(A-B)=\sin A\cos B-\cos A\sin B \] we get \[ \sin\left(x-\frac{\pi}{3}\right) = \sin x\cos\frac{\pi}{3}-\cos x\sin\frac{\pi}{3} \] \[ = \frac{1}{2}\sin x-\frac{\sqrt{3}}{2}\cos x \] \[ = \frac{1}{2}\left(\sin x-\sqrt{3}\cos x\right) \] Therefore, \[ \sin\left(x-\frac{\pi}{3}\right)\cos x = \frac{1}{2}\left(\sin x-\sqrt{3}\cos x\right)\cos x \]

Step 2: Rewrite the integral.
So, \[ I=\int \frac{dx}{\frac{1}{2}\left(\sin x-\sqrt{3}\cos x\right)\cos x} \] \[ I=2\int \frac{dx}{\left(\sin x-\sqrt{3}\cos x\right)\cos x} \] Divide numerator and denominator inside the expression by \(\cos^2x\).
Since, \[ \sin x=\tan x\cos x \] we have \[ \sin x-\sqrt{3}\cos x = \cos x(\tan x-\sqrt{3}) \] Thus, \[ \left(\sin x-\sqrt{3}\cos x\right)\cos x = \cos^2x(\tan x-\sqrt{3}) \] Therefore, \[ I=2\int \frac{dx}{\cos^2x(\tan x-\sqrt{3})} \] Since, \[ \frac{1}{\cos^2x}=\sec^2x \] we get \[ I=2\int \frac{\sec^2x}{\tan x-\sqrt{3}}\,dx \]

Step 3: Substitute \(t=\tan x-\sqrt{3}\).
Let \[ t=\tan x-\sqrt{3} \] Then, \[ dt=\sec^2x\,dx \] So, \[ I=2\int \frac{dt}{t} \] \[ I=2\log|t|+C \] Substituting back, \[ I=2\log|\tan x-\sqrt{3}|+C \] This can also be written as \[ I=2\log\left|\frac{\tan x-\sqrt{3}}{2}\right|+C \] because \[ 2\log 2 \] can be absorbed in the constant \(C\).

Step 4: Match with the options.
The matching option is \[ 2\log\left(\frac{\tan x-\sqrt{3}}{2}\right)+C \]

Step 5: Final conclusion.
Therefore, \[ \boxed{2\log\left(\frac{\tan x-\sqrt{3}}{2}\right)+C} \]