Question

\(\int _{\pi /5}^{3\pi /10}\frac{1}{1+\sqrt{\tan x}}dx=\cdots\)

• A. \(\frac{\pi }{10}\)
• B. \(\frac{\pi }{15}\)
• C. \(\frac{\pi }{20}\)
• D. \(\frac{\pi }{25}\)

Hint:

The limits add up to pi over 2, so use the property that replaces x by a plus b minus x and add the two forms.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The limits are \(\dfrac{\pi}{5}\) and \(\dfrac{3\pi}{10}\), whose sum is \(\dfrac{\pi}{2}\).
Use \(\int_a^b f(x)dx=\int_a^b f(a+b-x)dx\).

Step 2: Apply the Property
Replacing \(x\) by \(\dfrac{\pi}{2}-x\) turns \(\tan x\) into \(\cot x\):
\(I=\int\dfrac{\sqrt{\tan x}}{1+\sqrt{\tan x}}dx\).

Step 3: Add
\(2I=\int_{\pi/5}^{3\pi/10}1\,dx=\dfrac{3\pi}{10}-\dfrac{\pi}{5}=\dfrac{\pi}{10}\).

Step 4: Result
\(I=\dfrac{\pi}{20}\).

Final Answer:
The value of the integral is pi over 20. \[ \boxed{\text{(C) }\dfrac{\pi}{20}} \]