Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #100
Evaluate \(\int_0^\pi x\tan x/(\sec x\,\csc x)dx\), interpreting removable undefined points by continuous extension.
✅ Step-by-Step Solution
Final Answer: π²/4
- Where the original formula is defined, tan x/(sec x csc x)=sin²x. Its continuous extension on [0,π] is sin²x.
- Write I=∫₀^πx sin²x dx. Reflection gives I=∫₀^π(π−x)sin²x dx, so 2I=π∫₀^πsin²x dx.
- The last integral is π/2, by sin²x=(1−cos2x)/2. Hence I=π²/4.
- Therefore the integral equals π²/4.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Where the original formula is defined, tan x/(sec x csc x)=sin²x. Its continuous extension on [0,π] is sin²x.
- [1 mark] Write I=∫₀^πx sin²x dx. Reflection gives I=∫₀^π(π−x)sin²x dx, so 2I=π∫₀^πsin²x dx.
- [1 mark] The last integral is π/2, by sin²x=(1−cos2x)/2. Hence I=π²/4.
- [0.5 marks] Therefore the integral equals π²/4.
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