Evaluate the following Integrals.
`int sqrt(tan x) dx`

Answer

To evaluate the integral \( \int \sqrt{\tan x} \, dx \), we will use a substitution method. Let's go through the steps systematically.

### Step 1: Substitution
Let \( t = \sqrt{\tan x} \). Then, squaring both sides gives us:
\[ \tan x = t^2 \]

### Step 2: Differentiate
Now, we differentiate both sides with respect to \( x \):
\[ \sec^2 x \, dx = 2t \, dt \]
From this, we can express \( dx \) in terms of \( t \):
\[ dx = \frac{2t}{\sec^2 x} \, dt \]

### Step 3: Express \( \sec^2 x \) in terms of \( t \)
Using the identity \( \sec^2 x = 1 + \tan^2 x \), we can substitute \( \tan^2 x \) with \( t^4 \):
\[ \sec^2 x = 1 + t^4 \]
Thus, we can rewrite \( dx \):
\[ dx = \frac{2t}{1 + t^4} \, dt \]

### Step 4: Substitute into the integral
Now we substitute \( \sqrt{\tan x} \) and \( dx \) into the integral:
\[ \int \sqrt{\tan x} \, dx = \int t \cdot \frac{2t}{1 + t^4} \, dt = \int \frac{2t^2}{1 + t^4} \, dt \]

### Step 5: Split the integral
We can split the integral into two parts:
\[ \int \frac{2t^2}{1 + t^4} \, dt = \int \frac{2t^2 + 2 - 2}{1 + t^4} \, dt = \int \frac{2}{1 + t^4} \, dt - \int \frac{2}{1 + t^4} \, dt \]

### Step 6: Simplify the integral
This gives us:
\[ \int \frac{2}{1 + t^4} \, dt \]
To evaluate this integral, we can use the formula for the integral of the form \( \int \frac{dx}{a^2 + x^2} \) and some trigonometric identities.

### Step 7: Use a known integral
The integral \( \int \frac{1}{1 + t^4} \, dt \) can be evaluated using partial fractions or trigonometric substitution. The result is:
\[ \frac{1}{2} \tan^{-1}(t^2) + \frac{1}{2\sqrt{2}} \log \left| \frac{t^2 - 1}{t^2 + 1} \right| + C \]

### Step 8: Substitute back for \( t \)
Now, substituting back \( t = \sqrt{\tan x} \):
\[ \int \sqrt{\tan x} \, dx = \frac{1}{2} \tan^{-1}(\tan x) + \frac{1}{2\sqrt{2}} \log \left| \frac{\tan x - 1}{\tan x + 1} \right| + C \]

### Final Result
Thus, the final result for the integral is:
\[ \int \sqrt{\tan x} \, dx = \frac{1}{2} \tan^{-1}(\tan x) + \frac{1}{2\sqrt{2}} \log \left| \frac{\tan x - 1}{\tan x + 1} \right| + C \]

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