Example 32 - Evaluate integral  x sin x / 1 + cos2 x dx - Definate Integration by properties - P4

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  1. Chapter 7 Class 12 Integrals
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Example 32 Evaluate ∫_0^πœ‹β–’(π‘₯ 𝑠𝑖𝑛 π‘₯)/(1 + cos^2⁑π‘₯ ) 𝑑π‘₯ =(βˆ’πœ‹)/2 [tan^(βˆ’1)⁑(𝑑) ]_0^πœ‹ Putting t = cos x =(βˆ’πœ‹)/2 [tan^(βˆ’1)⁑(cos⁑π‘₯ ) ]_0^πœ‹ =(βˆ’πœ‹)/2 [tan^(βˆ’1)⁑〖[cosβ‘πœ‹ ]βˆ’tan^(βˆ’1)⁑[cos⁑0 ] γ€— ] =(βˆ’πœ‹)/2 [tan^(βˆ’1)⁑〖(βˆ’1)βˆ’tan^(βˆ’1)⁑(1) γ€— ] =(βˆ’πœ‹)/2 [βˆ’tan^(βˆ’1)⁑〖(1)βˆ’tan^(βˆ’1)⁑(1) γ€— ] =(βˆ’πœ‹)/2 [βˆ’2 tan^(βˆ’1)⁑(1) ] =πœ‹[tan^(βˆ’1)⁑(1) ] = πœ‹[πœ‹/4] =𝝅^𝟐/πŸ’

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