Q93 / 150 · MHT CET 2026 (17 April Shift 1) · Mathematics

The value of $\int_0^\pi \dfrac{1}{1 + 2^{\cos x}}\,dx$ is _____

  • Correct answer
    A
    $\dfrac{\pi}{2}$
  • B
    $\pi$
  • C
    $2\pi$
  • D
    $\dfrac{3\pi}{2}$

Solution

Let $I = \int_0^\pi \dfrac{1}{1 + 2^{\cos x}} \, dx$

Using the property $\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx$, we get:

$I = \int_0^\pi \dfrac{1}{1 + 2^{\cos(\pi - x)}} \, dx$

$I = \int_0^\pi \dfrac{1}{1 + 2^{-\cos x}} \, dx$

$I = \int_0^\pi \dfrac{2^{\cos x}}{2^{\cos x} + 1} \, dx$

Adding the two equations for $I$:

$2I = \int_0^\pi \left( \dfrac{1}{1 + 2^{\cos x}} + \dfrac{2^{\cos x}}{1 + 2^{\cos x}} \right) \, dx$

$2I = \int_0^\pi 1 \, dx$

$2I = \pi$

$I = \dfrac{\pi}{2}$

Answer: $\dfrac{\pi}{2}$

Hence, the correct answer is option: 'A'.

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