Question

Evaluate the integral: \[ \int \frac{\sqrt{\tan x}}{\sin x \cos x} \, dx \]

• A. \(2\sqrt{\tan x} + C\)
• B. \( \frac{2}{\sin^2 x} \)
• C. \( \frac{2}{\cos x} \)
• D. \( \frac{2}{\sin x} \)

Hint:

Remember to use standard trigonometric identities to simplify complex expressions and recognize common integrals.

Answer 1

The Correct Option is A

Step 1: Write the integral:
\[ \int \frac{\sqrt{\tan x}}{\sin x \cos x} \, dx \]

Step 2: Let \( t = \tan x \).
Then \( dt = \sec^2 x \, dx = \frac{1}{\cos^2 x} dx \)
\[ \Rightarrow dx = \cos^2 x \, dt \]

Step 3: Substitute into the integral:
\[ = \int \frac{\sqrt{t}}{\sin x \cos x} \cdot \cos^2 x \, dt \]

Step 4: Use identity \( \sin x = t \cos x \):
\[ \sin x \cos x = t \cos^2 x \]

Step 5: Simplify:
\[ = \int \frac{\sqrt{t} \cdot \cos^2 x}{t \cos^2 x} \, dt = \int \frac{1}{\sqrt{t}} \, dt \]

Step 6: Integrate:
\[ = \int t^{-1/2} dt = 2\sqrt{t} + C \]

Step 7: Substitute back \( t = \tan x \):
\[ = 2\sqrt{\tan x} + C \]

Final Answer: \(2\sqrt{\tan x} + C\)