Class 12 Integrals · Very Long Answer5 Marksdefinite_integration
Class 12 Maths · Integrals · Very Long Answer #176
Evaluate \(\int_{\pi/6}^{\pi/3}dx/(1+\sqrt{\cot x})\).
✅ Step-by-Step Solution
Final Answer: π/12
- Let I be the integral. The integrand is continuous on the closed interval, where cot x>0.
- Reflect x↦π/2−x to get I=∫_(π/6)^(π/3)dx/(1+√(tan x)).
- Because √(tan x)=1/√(cot x) here, the reflected integrand is √(cot x)/(1+√(cot x)).
- Adding gives 2I=∫_(π/6)^(π/3)1dx=π/3−π/6=π/6.
- Therefore the integral equals π/12.
📋 CBSE Stepwise Marking Scheme
- [1 mark] Let I be the integral. The integrand is continuous on the closed interval, where cot x>0.
- [1 mark] Reflect x↦π/2−x to get I=∫_(π/6)^(π/3)dx/(1+√(tan x)).
- [1 mark] Because √(tan x)=1/√(cot x) here, the reflected integrand is √(cot x)/(1+√(cot x)).
- [1 mark] Adding gives 2I=∫_(π/6)^(π/3)1dx=π/3−π/6=π/6.
- [1 mark] Therefore the integral equals π/12.
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