Slide34.JPG

Slide35.JPG
Slide36.JPG Slide37.JPG

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Example 30 Evaluate ∫_0^πœ‹β–’(π‘₯ 𝑠𝑖𝑛 π‘₯)/(1 + cos^2⁑π‘₯ ) 𝑑π‘₯ Let I=∫_0^πœ‹β–’(π‘₯ sin⁑π‘₯)/(1 + cos^2⁑π‘₯ ) 𝑑π‘₯ ∴ I=∫_0^πœ‹β–’((πœ‹ βˆ’ π‘₯) 𝑠𝑖𝑛(πœ‹ βˆ’ π‘₯))/(1 + cos^2⁑(πœ‹ βˆ’ π‘₯) ) 𝑑π‘₯ I=∫_0^πœ‹β–’γ€–((πœ‹ βˆ’ π‘₯) sin⁑π‘₯)/(1 + [βˆ’cos⁑π‘₯ ]^2 ) 𝑑π‘₯γ€— I=∫_0^πœ‹β–’γ€–(πœ‹ sin⁑〖π‘₯ βˆ’ π‘₯ sin⁑π‘₯ γ€—)/(1 + cos^2⁑π‘₯ ) 𝑑π‘₯γ€— Adding (1) and (2) i.e. (1) + (2) I +I=∫_0^πœ‹β–’(π‘₯ sin⁑π‘₯)/(1 + cos^2⁑π‘₯ ) 𝑑π‘₯+∫_0^πœ‹β–’(πœ‹ sin⁑〖π‘₯ βˆ’ π‘₯ 𝑠𝑖𝑛π‘₯γ€—)/(1 + cos^2⁑π‘₯ ) 𝑑π‘₯ 2I =∫_0^πœ‹β–’(π‘₯ sin⁑〖π‘₯ + πœ‹ sin⁑〖π‘₯ βˆ’ π‘₯ sin⁑π‘₯ γ€— γ€—)/(1 + cos^2⁑π‘₯ ) 𝑑π‘₯ 2I =∫_0^πœ‹β–’(πœ‹ sin⁑〖π‘₯ γ€—)/(1 + cos^2⁑π‘₯ ) 𝑑π‘₯ 2I =πœ‹βˆ«_0^πœ‹β–’sin⁑〖π‘₯ γ€—/(1 + cos^2⁑π‘₯ ) 𝑑π‘₯ ∴ I = πœ‹/2 ∫_0^πœ‹β–’γ€–sin⁑π‘₯/(1 + cos^2⁑π‘₯ ) 𝑑π‘₯γ€— Let cos π‘₯ = t Differentiate both sides w.r.t.π‘₯ – sin⁑〖π‘₯=𝑑𝑑/𝑑π‘₯γ€— 𝑑π‘₯=𝑑𝑑/(βˆ’sin⁑π‘₯ ) Putting the values of (cos⁑π‘₯ ) and dπ‘₯, we get I =πœ‹/2 ∫1_0^πœ‹β–’γ€–sin⁑π‘₯/(1 + 𝑑^2 ).𝑑π‘₯γ€— I = πœ‹/2 ∫1_0^πœ‹β–’γ€–sin⁑π‘₯/(1 + 𝑑^2 ) Γ— 𝑑𝑑/(βˆ’sin⁑π‘₯ )γ€— I = (βˆ’πœ‹)/2 ∫1_0^πœ‹β–’γ€–1/(1 + 𝑑^2 ) .𝑑𝑑〗 I = (βˆ’πœ‹)/2 ∫1_0^πœ‹β–’γ€–1/((1)^2 + (𝑑)^2 ) .𝑑𝑑〗 I=(βˆ’πœ‹)/2 [1/1 tan^(βˆ’1)⁑(𝑑/1) ]_π‘œ^πœ‹ =(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’

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.