Find `int(sqrt(cotx)+sqrt(tanx))dx`

Options

A. `sqrt(2)sin^(-1)(sin x - cos x)`
B. `(pi)/(2)-sqrt(2) cos^(-1)(sin x- cos x)`
C. `sqrt(2) tan^(-1)((tan x-1)/(sqrt(2)sqrt(tan x)))`
D. None of these

Answer

To solve the integral \( \int (\sqrt{\cot x} + \sqrt{\tan x}) \, dx \), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Rewrite the Trigonometric Functions**:
We know that:
\[ \cot x = \frac{\cos x}{\sin x} \quad \text{and} \quad \tan x = \frac{\sin x}{\cos x} \]
Therefore, we can express the integral as:
\[ \int \left( \sqrt{\frac{\cos x}{\sin x}} + \sqrt{\frac{\sin x}{\cos x}} \right) \, dx \]

2. **Combine the Terms**:
We can combine the square roots:
\[ \int \left( \frac{\sqrt{\cos x}}{\sqrt{\sin x}} + \frac{\sqrt{\sin x}}{\sqrt{\cos x}} \right) \, dx = \int \left( \frac{\sqrt{\cos x} \cdot \sqrt{\cos x} + \sqrt{\sin x} \cdot \sqrt{\sin x}}{\sqrt{\sin x} \cdot \sqrt{\cos x}} \right) \, dx \]
This simplifies to:
\[ \int \frac{\sqrt{\sin x \cos x} (\sqrt{\cos x} + \sqrt{\sin x})}{\sqrt{\sin x \cos x}} \, dx \]

3. **Finding a Common Denominator**:
Taking the common denominator:
\[ \int \frac{\sqrt{\sin x \cos x} (\sqrt{\cos x} + \sqrt{\sin x})}{\sqrt{\sin x \cos x}} \, dx = \int \frac{\sqrt{\sin x} + \sqrt{\cos x}}{\sqrt{\sin x \cos x}} \, dx \]

4. **Using the Half-Angle Identity**:
We know that:
\[ 2 \sin x \cos x = \sin 2x \]
Thus, we can express the integral as:
\[ \int \frac{\sqrt{2}}{\sqrt{\sin 2x}} (\sqrt{\sin x} + \sqrt{\cos x}) \, dx \]

5. **Substituting for Simplification**:
Let \( t = \sin x - \cos x \). Then, we differentiate:
\[ dt = \cos x \, dx - (-\sin x \, dx) = (\cos x + \sin x) \, dx \]
So, we have:
\[ dx = \frac{dt}{\cos x + \sin x} \]

6. **Substituting Back into the Integral**:
Substitute \( dx \) into the integral:
\[ \int \frac{\sqrt{2} (\sqrt{\sin x} + \sqrt{\cos x})}{\sqrt{1 - t^2}} \cdot \frac{dt}{\cos x + \sin x} \]
The \( \cos x + \sin x \) cancels out.

7. **Final Integration**:
The integral simplifies to:
\[ \sqrt{2} \int \frac{dt}{\sqrt{1 - t^2}} = \sqrt{2} \sin^{-1}(t) + C \]
Substituting back \( t = \sin x - \cos x \):
\[ = \sqrt{2} \sin^{-1}(\sin x - \cos x) + C \]

### Final Answer:
\[ \int (\sqrt{\cot x} + \sqrt{\tan x}) \, dx = \sqrt{2} \sin^{-1}(\sin x - \cos x) + C \]