Q88 / 180 · COMEDK 2025 (10 May Shift 2) · Mathematics

$\int_0^{\dfrac{\pi}{2}} \log \left(\dfrac{5+4 \sin x}{5+4 \cos x}\right) d x=$

  • A
    2
  • B
    -2
  • Correct answer
    C
    0
  • D
    $\dfrac{3}{4}$

Solution

Let $I = \int_{0}^{\dfrac{\pi}{2}} \log \left( \dfrac{5 + 4 \sin x}{5 + 4 \cos x} \right) dx$.

Using the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a - x) dx$, we have:

$I = \int_{0}^{\dfrac{\pi}{2}} \log \left( \dfrac{5 + 4 \sin(\dfrac{\pi}{2} - x)}{5 + 4 \cos(\dfrac{\pi}{2} - x)} \right) dx$

$I = \int_{0}^{\dfrac{\pi}{2}} \log \left( \dfrac{5 + 4 \cos x}{5 + 4 \sin x} \right) dx$

$I = \int_{0}^{\dfrac{\pi}{2}} \log \left( \left( \dfrac{5 + 4 \sin x}{5 + 4 \cos x} \right)^{-1} \right) dx$

$I = - \int_{0}^{\dfrac{\pi}{2}} \log \left( \dfrac{5 + 4 \sin x}{5 + 4 \cos x} \right) dx$

$I = -I$

$2I = 0 \Rightarrow I = 0$.

Answer: 0

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

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