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Ex 7.3, 9 - Integrate cos x / 1 + cos x - Class 12 CBSE - Ex 7.3

Ex 7.3, 9 - Chapter 7 Class 12 Integrals - Part 2

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Ex 7.3, 9 Integrate (π‘π‘œπ‘  π‘₯)/(1 + π‘π‘œπ‘  π‘₯) ∫1β–’γ€–(π‘π‘œπ‘  π‘₯)/(1 + π‘π‘œπ‘  π‘₯) " " 𝑑π‘₯γ€— = ∫1β–’((cos⁑π‘₯ + 1 βˆ’ 1)/(1 + cos⁑π‘₯ )) 𝑑π‘₯ =∫1β–’((1 + cos⁑π‘₯ βˆ’ 1)/(1 + cos⁑π‘₯ )) 𝑑π‘₯ =∫1β–’((1 + cos⁑π‘₯)/(1 + cos⁑π‘₯ ) βˆ’ 1/(1 + cos⁑π‘₯ )) 𝑑π‘₯ =∫1β–’γ€–1βˆ’1/(1 + cos⁑π‘₯ )γ€— 𝑑π‘₯ =∫1β–’1 𝑑π‘₯βˆ’βˆ«1β–’πŸ/(𝟏 + 𝒄𝒐𝒔⁑𝒙 ) 𝑑π‘₯ =∫1β–’1 𝑑π‘₯βˆ’βˆ«1β–’1/(𝟐 〖𝒄𝒐𝒔〗^πŸβ‘γ€–π’™/πŸγ€— ) 𝑑π‘₯ =∫1β–’1 𝑑π‘₯βˆ’βˆ«1β–’1/2 sec^2⁑〖π‘₯/2γ€— 𝑑π‘₯ =∫1β–’1 𝑑π‘₯βˆ’1/2 ∫1β–’sec^2⁑〖π‘₯/2γ€— 𝑑π‘₯ =π‘₯βˆ’ 1/2 γ€–tan 〗⁑〖π‘₯/2γ€—/(1/2) +𝐢 =π‘₯βˆ’ 2/2 γ€–tan 〗⁑〖π‘₯/2γ€— +𝐢 =π’™βˆ’ γ€–π­πšπ§ 〗⁑〖𝒙/πŸγ€— +π‘ͺ ∫1β–’sec^2⁑(π‘Žπ‘₯+𝑏) 𝑑π‘₯=π‘‘π‘Žπ‘›β‘(π‘Žπ‘₯ + 𝑏)/π‘Ž +𝐢 We know that cos 2πœƒ=2 cos^2β‘γ€–πœƒβˆ’1γ€— cos⁑2πœƒ+1=2 cos^2β‘πœƒ Replacing πœƒ by π‘₯/2 cos⁑2(π‘₯/2)+1=2 cos^2⁑〖π‘₯/2γ€— cos⁑π‘₯+1=2 cos^2⁑〖π‘₯/2γ€—

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