Check sibling questions
Chapter 7 Class 12 Integrals
Concept wise

Misc 25 - Evaluate definite integral ex (1-sin x / 1-cos x)

Misc 25 - Chapter 7 Class 12 Integrals - Part 2
Misc 25 - Chapter 7 Class 12 Integrals - Part 3

Maths Crash Course - Live lectures + all videos + Real time Doubt solving!


Transcript

Misc 25 Evaluate the definite integral ∫_(πœ‹/2)^πœ‹β–’γ€–e^π‘₯⁑((1 βˆ’sin⁑π‘₯)/(1 βˆ’cos⁑π‘₯ )) 𝑑π‘₯γ€— ∫_(πœ‹/2)^πœ‹β–’γ€–e^π‘₯⁑((1 βˆ’ sin⁑π‘₯)/(1 βˆ’γ€– cos〗⁑π‘₯ )) 𝑑π‘₯γ€— = ∫_(πœ‹/2)^πœ‹β–’γ€–e^π‘₯⁑((1 )/(1 βˆ’ cos⁑π‘₯ )βˆ’sin⁑π‘₯/(1 βˆ’ cos⁑π‘₯ )) 𝑑π‘₯γ€— = ∫_(πœ‹/2)^πœ‹β–’γ€–e^π‘₯⁑((βˆ’sin⁑π‘₯)/(1 βˆ’ cos⁑π‘₯ )+(1 )/(1 βˆ’ cos⁑π‘₯ )) 𝑑π‘₯γ€— Let f(x) = (βˆ’sin⁑π‘₯)/(1 βˆ’ cos⁑π‘₯ ) f’(x) = βˆ’[(cos⁑π‘₯ (1 βˆ’ cos⁑〖π‘₯) βˆ’ (sin⁑〖π‘₯) (sin⁑〖π‘₯)γ€— γ€— γ€—)/((1 βˆ’ γ€–cos⁑π‘₯)γ€—^2 )] =βˆ’[(cos⁑〖π‘₯ βˆ’γ€— γ€–π‘π‘œπ‘ γ€—^2 π‘₯ βˆ’ 〖𝑠𝑖𝑛〗^2 π‘₯)/((1 βˆ’ γ€–cos π‘₯⁑)γ€—^2 )] = βˆ’(cos⁑〖π‘₯ βˆ’ (γ€–π‘π‘œπ‘ γ€—^2 π‘₯ + 〖𝑠𝑖𝑛〗^2 π‘₯)γ€—/((1 βˆ’ γ€–π‘π‘œπ‘  π‘₯⁑)γ€—^2 )) = βˆ’ (cos⁑〖π‘₯ βˆ’ 1γ€—/γ€–(1 βˆ’γ€– cos〗⁑〖π‘₯)γ€—γ€—^2 ) = ( 1 βˆ’ cos⁑〖π‘₯ γ€—)/γ€–(1 βˆ’ cos⁑〖π‘₯)γ€—γ€—^2 = 1/(1 βˆ’ cos⁑π‘₯ ) Hence, the given integration is of form, ∫1▒〖𝑒^π‘₯ (𝑓(π‘₯)+𝑓^β€² (π‘₯)) 𝑑π‘₯γ€—= 𝑒^π‘₯ 𝑓(π‘₯) Where f(x) = (βˆ’sin⁑π‘₯)/(1 βˆ’ cos⁑π‘₯ ) and f’(x) = 1/(1 βˆ’ cos⁑π‘₯ ) Hence, ∫_(πœ‹/2)^πœ‹β–’γ€–e^π‘₯⁑((1 βˆ’ sin⁑π‘₯)/(1 βˆ’γ€– cos〗⁑π‘₯ )) 𝑑π‘₯γ€— = ∫_(πœ‹/2)^πœ‹β–’γ€–e^π‘₯⁑((1 )/(1 βˆ’ cos⁑π‘₯ )βˆ’sin⁑π‘₯/(1 βˆ’ cos⁑π‘₯ )) 𝑑π‘₯γ€— = [𝑒^π‘₯ ((βˆ’sin⁑π‘₯)/(1 βˆ’cos⁑π‘₯ ))]_(πœ‹/2)^πœ‹ = ((𝑒^π‘₯ sin⁑π‘₯)/cos⁑〖π‘₯ βˆ’1γ€— )_(πœ‹/2)^πœ‹ Putting limits = (𝑒^πœ‹ sinβ‘πœ‹)/cosβ‘γ€–πœ‹ βˆ’1γ€— βˆ’ (𝑒^(πœ‹/2) sinβ‘γ€–πœ‹/2γ€—)/cos⁑〖 πœ‹/2 βˆ’1γ€— = (𝑒^πœ‹ Γ— (0))/(βˆ’1 βˆ’1)βˆ’ (𝑒^(πœ‹/2) (1))/(0 βˆ’ 1) = 𝒆^(𝝅/𝟐)

Ask a doubt (live)
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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.