NCERT Class 12 · Ch 7 Exercise 7.10 · Q16
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.10 Question 16
Evaluate \(\int\limits_{0}^{\pi} \log{\left(\cos{\left(x \right)} + 1 \right)}\, dx\) using definite-integral properties.
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: \(- \pi \log{\left(2 \right)}\)
- Use 1+cos x=2cos²(x/2) and t=x/2: I=πlog 2+4J, J=∫₀^(π/2)log cos t dt.
- Reflection gives J=∫log sin t dt. Adding these forms and using sin 2t=2sin t cos t yields 2J=J−(π/2)log 2, so J=−(π/2)log 2.
- Hence I=−πlog 2. The singularity at x=π is logarithmic and integrable.
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