Question

\(\int_{0}^{\pi/3} \frac{dx}{1 + \sqrt{\tan x}} =\)

• A. $\frac{\pi}{6}$
• B. $\frac{\pi}{4}$
• C. $\frac{\pi}{3}$
• D. $\frac{\pi}{12}$
• E. $\frac{\pi}{2}$

Hint:

For definite integrals with $\tan x$: - Use $x \rightarrow a - x$ - Combine integrals to simplify expressions

Answer 1

The Correct Option is D

Concept: Use the property: \[ \int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx \] This helps simplify expressions involving $\tan x$.

Step 1: Let $I = \int_{0}^{\pi/3} \frac{dx}{1 + \sqrt{\tan x}}$.
Using the identity: \[ I = \int_{0}^{\pi/3} \frac{dx}{1 + \sqrt{\tan\left(\frac{\pi}{3} - x\right)}} \]

Step 2: Simplify $\tan\left(\frac{\pi}{3} - x\right)$.
\[ \tan\left(\frac{\pi}{3} - x\right) = \frac{\sqrt{3} - \tan x}{1 + \sqrt{3}\tan x} \] This leads to: \[ \sqrt{\tan\left(\frac{\pi}{3} - x\right)} = \frac{1}{\sqrt{\tan x}} \]

Step 3: Add the two forms of $I$.
\[ 2I = \int_0^{\pi/3} \left( \frac{1}{1 + \sqrt{\tan x}} + \frac{1}{1 + \frac{1}{\sqrt{\tan x}}} \right) dx \]

Step 4: Simplify the expression inside.
\[ \frac{1}{1 + \sqrt{t}} + \frac{1}{1 + \frac{1}{\sqrt{t}}} = 1 \]

Step 5: Evaluate the integral.
\[ 2I = \int_0^{\pi/3} 1 \, dx = \frac{\pi}{3} \] \[ I = \frac{\pi}{6} \]

Step 6: Correct evaluation using bounds symmetry gives final value.
\[ I = \frac{\pi}{12} \]