Question:

Step 1: Simplify the expression inside the square root. First, expand the term inside the square root and then use the identity $1 + \cot^2 x = \csc^2 x$. $$ 1 + 2\cot x(\cot x + \csc x) = 1 + 2\cot^2 x + 2\cot x \csc x $$ $$ = (1 + \cot^2 x) + \cot^2 x + 2\cot x \csc x $$ $$ = \csc^2 x + \cot^2 x + 2\cot x \csc x $$ Recognize this as a perfect square of the form $(a+b)^2 = a^2 + b^2 + 2ab$. $$ = (\csc x + \cot x)^2 $$ Step 2: Rewrite the integral with the simplified expression. Substitute the simplified expression back into the integral. Since we are dealing with a square root of a square, we take the absolute value, but in typical JEE problems, we assume the domain where $\csc x + \cot x$ is positive. $$ \int \sqrt{(\csc x + \cot x)^2} \, dx = \int (\csc x + \cot x) \, dx $$ Step 3: Express the integrand in terms of sine and cosine functions. Rewrite $\csc x$ and $\cot x$ using their definitions $\csc x = \frac{1}{\sin x}$ and $\cot x = \frac{\cos x}{\sin x}$. $$ \int \left(\frac{1}{\sin x} + \frac{\cos x}{\sin x}\right) \, dx = \int \frac{1 + \cos x}{\sin x} \, dx $$ Step 4: Use half-angle identities to further simplify the integrand. Apply the half-angle identities $1 + \cos x = 2\cos^2\left(\frac{x}{2}\right)$ and $\sin x = 2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$. $$ \int \frac{2\cos^2\left(\frac{x}{2}\right)}{2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)} \, dx = \int \frac{\cos\left(\frac{x}{2}\right)}{\sin\left(\frac{x}{2}\right)} \, dx $$ $$ = \int \cot\left(\frac{x}{2}\right) \, dx $$ Step 5: Integrate the simplified expression. Recall the standard integral $\int \cot(ax+b) \, dx = \frac{1}{a}\ln|\sin(ax+b)| + c$. For this case, $a = \frac{1}{2}$. $$ \int \cot\left(\frac{x}{2}\right) \, dx = \frac{1}{1/2}\ln\left|\sin\left(\frac{x}{2}\right)\right| + c $$ $$ = 2\ln\left|\sin\left(\frac{x}{2}\right)\right| + c $$ Step 6: State the final answer and match with the given options. The integral of $\sqrt{1 + 2\cot x(\cot x + \csc x)}$ w.r.t. $x$ is $2\ln\left|\sin\left(\frac{x}{2}\right)\right| + c$. Comparing this with the given options, it matches Option 2. The final answer is $\boxed{2\ln\sin\frac{x}{2}+c}$.