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NCERT Class 12 · Ch 7 Exercise 7.10 · Q10

NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.10 Question 10

Evaluate \(\int\limits_{0}^{\frac{\pi}{2}} \left(2 \log{\left(\sin{\left(x \right)} \right)} - \log{\left(\sin{\left(2 x \right)} \right)}\right)\, dx\) using definite-integral properties.

💡 Concept Hints

  1. Identify the domain and the inner and outer functions, or the relevant one-sided limits.
  2. Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.

✅ Step-by-Step Solution

Final Answer: \(- \frac{\pi \log{\left(2 \right)}}{2}\)
  1. Use sin 2x=2sin x cos x to rewrite the integrand as log sin x−log cos x−log 2.
  2. The first two integrals are equal by reflection x↦π/2−x and cancel. The remaining constant integral is −(π/2)log 2. The logarithmic endpoint singularities are integrable.

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