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

Answer

To solve the integral \(\int \frac{\sec^2 x}{\sqrt{\tan x}} \, dx\), we can use the substitution method. Here are the steps:

### Step 1: Substitution
Let \( t = \tan x \). Then, we differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = \sec^2 x \quad \Rightarrow \quad dt = \sec^2 x \, dx \]
From this, we can express \( dx \) in terms of \( dt \):
\[ dx = \frac{dt}{\sec^2 x} \]

### Step 2: Rewrite the Integral
Substituting \( t = \tan x \) into the integral, we have:
\[ \int \frac{\sec^2 x}{\sqrt{\tan x}} \, dx = \int \frac{\sec^2 x}{\sqrt{t}} \cdot \frac{dt}{\sec^2 x} \]
The \(\sec^2 x\) terms cancel out:
\[ = \int \frac{1}{\sqrt{t}} \, dt \]

### Step 3: Integrate
Now, we can integrate \(\frac{1}{\sqrt{t}}\):
\[ \int \frac{1}{\sqrt{t}} \, dt = 2\sqrt{t} + C \]

### Step 4: Substitute Back
Now, we substitute back \( t = \tan x \):
\[ 2\sqrt{t} + C = 2\sqrt{\tan x} + C \]

### 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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