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: Pair the integrand
Define $g(x)=\dfrac{1}{1+2^{\cos x}}$. Then $g(x)+g(\pi-x)=\dfrac{1}{1+2^{\cos x}}+\dfrac{2^{\cos x}}{2^{\cos x}+1}=1$.

Step 2: Integrate
$\int_0^\pi g(x)dx=\tfrac12\int_0^\pi[g(x)+g(\pi-x)]dx=\tfrac12\pi$, option (A).

Final Answer:
The integral equals $\pi/2$, option (A). \[ \boxed{\dfrac\pi2} \]