`int(sqrt(tanx)+sqrt(cotx))dx` is equal to

Options

A. `sqrt(2)sin^(-1)(sinx+cosx)+C`
B. `sqrt(2)cos^(-1)(sinx-cosx)+C`
C. `sqrt(2)sin^(-1)(sinx-cosx)+C`
D. `sqrt(2)cos^(-1)(sinx+cosx)+C`

Answer

To solve the integral \( \int (\sqrt{\tan x} + \sqrt{\cot x}) \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral in terms of sine and cosine:
\[ \int \left( \sqrt{\frac{\sin x}{\cos x}} + \sqrt{\frac{\cos x}{\sin x}} \right) \, dx = \int \left( \frac{\sqrt{\sin x}}{\sqrt{\cos x}} + \frac{\sqrt{\cos x}}{\sqrt{\sin x}} \right) \, dx \]

### Step 2: Combine the Terms
Next, we can combine the terms under 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 Integral
Now, we can express \(\sin x + \cos x\) in a more manageable form:
\[ = \int \frac{\sqrt{2} \cdot \left( \frac{1}{\sqrt{2}}(\sin x + \cos x) \right)}{\sqrt{\sin x \cos x}} \, dx \]
Using the identity \( \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \):
\[ = \sqrt{2} \int \frac{\sin\left(x + \frac{\pi}{4}\right)}{\sqrt{\sin x \cos x}} \, dx \]

### Step 4: Use Substitution
Let \( t = \sin x - \cos x \). Then, differentiate:
\[ dt = (\cos x + \sin x) \, dx \]
This implies:
\[ dx = \frac{dt}{\cos x + \sin x} \]

### Step 5: Express in Terms of \( t \)
Now we need to express \( \sin 2x \) in terms of \( t \):
\[ \sin^2 x + \cos^2 x - 2 \sin x \cos x = 1 - 2 \sin x \cos x = 1 - \frac{1 - t^2}{2} \]
Thus, we have:
\[ 1 - t^2 = \sin 2x \]

### Step 6: Substitute Back into the Integral
Now we can substitute back into the integral:
\[ = \sqrt{2} \int \frac{dt}{\sqrt{1 - t^2}} \]

### Step 7: Integrate
The integral of \( \frac{1}{\sqrt{1 - t^2}} \) is:
\[ = \sqrt{2} \sin^{-1}(t) + C \]

### Step 8: Substitute \( t \) Back
Finally, substituting back for \( t \):
\[ = \sqrt{2} \sin^{-1}(\sin x - \cos x) + C \]

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