Question

Evaluate the definite integral. $$ \int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} \frac{x}{1+\sin x} \, dx = $$

• A. \(\pi(\sqrt{3} - 2)\)
• B. \(\pi(2 - \sqrt{3})\)
• C. \(\pi(\sqrt{3} + 2)\)
• D. \(\pi/2(2 - \sqrt{3})\)
• E. None of these

Hint:

For any definite integral, the three essential parts are:
• integrand,
• variable,
• limits. e} If even one of these is missing, the answer cannot be found uniquely.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Use \(\int_a^b f(x) = \int_a^b f(a+b-x)\).
Step 2: Key Formula or Approach:
Add two versions of the integral to remove the numerator \(x\).
Step 3: Detailed Explanation:
\(2I = \int \frac{\pi}{1+\sin x} dx = \pi [\tan x - \sec x]\) from \(\pi/3\) to \(2\pi/3\).
Calculation leads to \(2I = \pi(4 - 2\sqrt{3}) \implies I = \pi(2-\sqrt{3})\).
Step 4: Final Answer:
Result is \(\pi(2-\sqrt{3})\).