Evaluate : `int ( sqrt( tan x) - sqrt( cot x)) dx `

Answer

To evaluate the integral \( \int \left( \sqrt{\tan x} - \sqrt{\cot x} \right) \, dx \), we can follow these steps:

### Step 1: Rewrite the integrand
We start by rewriting \( \sqrt{\tan x} \) and \( \sqrt{\cot x} \) in terms of sine and cosine:
\[ \sqrt{\tan x} = \sqrt{\frac{\sin x}{\cos x}} = \frac{\sqrt{\sin x}}{\sqrt{\cos x}} \]
\[ \sqrt{\cot x} = \sqrt{\frac{\cos x}{\sin x}} = \frac{\sqrt{\cos x}}{\sqrt{\sin x}} \]
Thus, we can rewrite the integral as:
\[ \int \left( \frac{\sqrt{\sin x}}{\sqrt{\cos x}} - \frac{\sqrt{\cos x}}{\sqrt{\sin x}} \right) \, dx \]

### Step 2: Find a common denominator
To combine the two fractions, we find a common denominator:
\[ \int \left( \frac{\sqrt{\sin x} \cdot \sqrt{\sin x} - \sqrt{\cos x} \cdot \sqrt{\cos x}}{\sqrt{\sin x \cos x}} \right) \, dx = \int \frac{\sin x - \cos x}{\sqrt{\sin x \cos x}} \, dx \]

### Step 3: Simplify the expression
Now we can express the integral as:
\[ \int \frac{\sin x - \cos x}{\sqrt{\sin x \cos x}} \, dx \]

### Step 4: Substitute variables
Let \( t = \sin x + \cos x \). Then, we differentiate:
\[ dt = (\cos x - \sin x) \, dx \]
This means:
\[ dx = \frac{dt}{\cos x - \sin x} \]

### Step 5: Substitute back into the integral
Now we can substitute back into the integral:
\[ \int \frac{\sin x - \cos x}{\sqrt{\sin x \cos x}} \cdot \frac{dt}{\cos x - \sin x} \]
This simplifies to:
\[ -\int \frac{dt}{\sqrt{\sin x \cos x}} \]

### Step 6: Use trigonometric identities
Using the identity \( \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \) and \( \sin x \cos x = \frac{1}{2} \sin(2x) \), we can further simplify the integral.

### Step 7: Solve the integral
The integral can now be solved using standard integral formulas. The final result will involve logarithmic and square root terms based on the substitutions made.

### Final Answer
The final answer, after substituting back and simplifying, will be:
\[ -\sqrt{2} \ln\left(\sin x + \cos x + \sqrt{(\sin x + \cos x)^2 - 1}\right) + C \]