Question

The value of \(\int _0^π\frac{1}{1+2^{cosx}}\,dx\) is _____

• A. \(\frac{π}{2}\)
• B. \(π\)
• C. \(2π\)
• D. \(\frac{3π}{2}\)

Hint:

Use \(\int_0^af(x)dx=\int_0^af(a-x)dx\) and add.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Replace \(x\) by \(\pi-x\). Since \(\cos(\pi-x)=-\cos x\), the integrand changes form.

Step 2: Key Formula or Approach
\[ I=\int_0^\pi\frac{dx}{1+2^{\cos x}}=\int_0^\pi\frac{dx}{1+2^{-\cos x}}=\int_0^\pi\frac{2^{\cos x}dx}{2^{\cos x}+1} \]

Step 3: Detailed Explanation
Add the two forms:
\[ 2I=\int_0^\pi\frac{1+2^{\cos x}}{1+2^{\cos x}}dx=\int_0^\pi dx=\pi \]
\[ I=\frac\pi2 \]

Final Answer:
The value is \(\pi/2\), option (A). \[ \boxed{\dfrac\pi2\ \text{(A)}} \]