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NCERT Class 12 · Ch 7 Exercise 7.10 · Q8

NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.10 Question 8

Evaluate \(\int\limits_{0}^{\frac{\pi}{4}} \log{\left(\tan{\left(x \right)} + 1 \right)}\, dx\) 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: \(\frac{\pi \log{\left(2 \right)}}{8}\)
  1. Reflect x↦π/4−x; tan(π/4−x)=(1−tan x)/(1+tan x).
  2. Thus log(1+tan x)+log(1+tan(π/4−x))=log 2. Integrating gives 2I=(π/4)log 2.

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All Questions in Integrals — Exercise 7.10

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