Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #66
Evaluate \(\int_0^{\pi/4}\ln(1+\tan x)dx\).
✅ Step-by-Step Solution
Final Answer: (π/8)ln2
- Let I be the integral. Reflect x↦π/4−x without changing its value.
- Since 1+tan(π/4−x)=2/(1+tan x), the reflected integrand is ln2−ln(1+tan x).
- Adding the two representations gives 2I=∫₀^(π/4)ln2 dx=(π/4)ln2.
- Therefore the integral equals (π/8)ln2.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let I be the integral. Reflect x↦π/4−x without changing its value.
- [1 mark] Since 1+tan(π/4−x)=2/(1+tan x), the reflected integrand is ln2−ln(1+tan x).
- [1 mark] Adding the two representations gives 2I=∫₀^(π/4)ln2 dx=(π/4)ln2.
- [0.5 marks] Therefore the integral equals (π/8)ln2.
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