`int(sec^(2)x)/(sqrt(tanx))dx`

Answer

To solve the integral \(\int \frac{\sec^2 x}{\sqrt{\tan x}} \, dx\), we will use substitution and integration techniques. Here’s a step-by-step solution:

### Step 1: Set up the integral
Let
\[ I = \int \frac{\sec^2 x}{\sqrt{\tan x}} \, dx \]

### Step 2: Use substitution
We will substitute \( t = \tan x \). Then, we differentiate both sides:
\[ \frac{dt}{dx} = \sec^2 x \quad \Rightarrow \quad dt = \sec^2 x \, dx \]
This means we can replace \( \sec^2 x \, dx \) with \( dt \).

### Step 3: Substitute in the integral
Now, substituting \( t \) into the integral, we have:
\[ I = \int \frac{1}{\sqrt{t}} \, dt \]

### Step 4: Rewrite the integral
The integral can be rewritten as:
\[ I = \int t^{-\frac{1}{2}} \, dt \]

### Step 5: Integrate
Using the power rule for integration, we find:
\[ I = \frac{t^{-\frac{1}{2} + 1}}{-\frac{1}{2} + 1} + C = \frac{t^{\frac{1}{2}}}{\frac{1}{2}} + C = 2\sqrt{t} + C \]

### Step 6: Substitute back
Now, we substitute back \( t = \tan x \):
\[ I = 2\sqrt{\tan x} + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{\sec^2 x}{\sqrt{\tan x}} \, dx = 2\sqrt{\tan x} + C \]

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