Question:

Step 1: Simplify the expression inside the square root. We start by expanding the term inside the square root and then use the trigonometric identity $1 + \cot^2 x = \csc^2 x$. $$1 + 2\cot x(\cot x + \csc x) = 1 + 2\cot^2 x + 2\cot x \csc x$$ $$ = (1 + \cot^2 x) + \cot^2 x + 2\cot x \csc x$$ $$ = \csc^2 x + \cot^2 x + 2\cot x \csc x$$ Step 2: Factor the simplified expression as a perfect square. The expression obtained in Step 1 is in the form $a^2 + b^2 + 2ab$, which is equal to $(a+b)^2$. $$ \csc^2 x + \cot^2 x + 2\cot x \csc x = (\csc x + \cot x)^2 $$ Step 3: Substitute the simplified expression back into the integral. Now, we replace the original expression inside the square root with its simplified form. We assume $\csc x + \cot x$ is non-negative in the domain of integration, so $\sqrt{(\csc x + \cot x)^2} = \csc x + \cot x$. $$ \int \sqrt{1 + 2\cot x(\cot x + \csc x)} dx = \int \sqrt{(\csc x + \cot x)^2} dx $$ $$ = \int (\csc x + \cot x) dx $$ Step 4: Rewrite the integrand in terms of sine and cosine. Express $\csc x$ as $\frac{1}{\sin x}$ and $\cot x$ as $\frac{\cos x}{\sin x}$ to facilitate further simplification. $$ \int \left(\frac{1}{\sin x} + \frac{\cos x}{\sin x}\right) dx = \int \frac{1+\cos x}{\sin x} dx $$ Step 5: Apply half-angle trigonometric identities. Use the identities $1+\cos x = 2\cos^2\left(\frac{x}{2}\right)$ and $\sin x = 2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$ to transform the integrand. $$ \int \frac{2\cos^2\left(\frac{x}{2}\right)}{2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)} dx $$ Step 6: Simplify the integrand and perform the integration. Cancel out common terms and simplify the expression to a basic trigonometric function, then integrate. $$ \int \frac{\cos\left(\frac{x}{2}\right)}{\sin\left(\frac{x}{2}\right)} dx = \int \cot\left(\frac{x}{2}\right) dx $$ The integral of $\cot u$ is $\ln|\sin u| + C$. For $\cot(ax+b)$, the integral is $\frac{1}{a}\ln|\sin(ax+b)| + C$. Here, $a = \frac{1}{2}$. $$ \int \cot\left(\frac{x}{2}\right) dx = \frac{1}{1/2}\ln\left|\sin\left(\frac{x}{2}\right)\right| + c $$ $$ = 2\ln\left|\sin\left(\frac{x}{2}\right)\right| + c $$ The final answer is $\boxed{2\ln\sin\frac{x}{2}+c}$.