Chapter 7 Class 12 Integrals
Concept wise

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Ex 7.10, 1 By using the properties of definite integrals, evaluate the integrals : ∫_0^(πœ‹/2)β–’γ€–cos^2⁑π‘₯ 𝑑π‘₯γ€— Let 𝐈=∫_𝟎^(𝝅/𝟐)▒〖〖𝒄𝒐𝒔〗^πŸβ‘π’™ 𝒅𝒙〗 I=∫_𝟎^(𝝅/𝟐)β–’γ€–γ€–πœπ¨π¬γ€—^𝟐⁑ (𝝅/πŸβˆ’π’™)𝒅𝒙〗 I= ∫_𝟎^((𝝅 )/𝟐)▒〖〖𝐬𝐒𝐧〗^𝟐 𝒙〗⁑𝒅𝒙 Adding (1) and (2) I+I= ∫_0^(πœ‹/2)β–’γ€–cos^2⁑π‘₯ 𝑑π‘₯γ€— + ∫_0^((πœ‹ )/2)β–’γ€–sin^2 π‘₯〗⁑𝑑π‘₯ 2I= ∫_0^((πœ‹ )/2)β–’(cos^2⁑〖π‘₯+sin^2⁑π‘₯ γ€— )⁑𝑑π‘₯ 𝟐𝐈 =∫_𝟎^((𝝅 )/𝟐)β–’γ€–πŸ .〗⁑𝒅𝒙 2I=[π‘₯]_0^(πœ‹/2) 2I =πœ‹/2βˆ’0 2I =πœ‹/2 𝐈=𝝅/πŸ’ Evaluate: ∫_0^πœ‹β–’γ€–π‘’^cos⁑π‘₯ /(𝑒^cos⁑π‘₯ + 𝑒^γ€–βˆ’cos〗⁑π‘₯ ) 𝑑π‘₯γ€— Let I=∫_0^πœ‹β–’γ€–π‘’^cos⁑π‘₯ /(𝑒^cos⁑π‘₯ + 𝑒^γ€–βˆ’cos〗⁑π‘₯ ) 𝑑π‘₯γ€— " " I= ∫_0^πœ‹β–’γ€–π‘’^cos⁑〖(πœ‹ βˆ’ π‘₯)γ€— /(𝑒^cos⁑〖(πœ‹ βˆ’ π‘₯)γ€— + 𝑒^γ€–βˆ’cos〗⁑〖(πœ‹ βˆ’ π‘₯)γ€— ) 𝑑π‘₯γ€— " " I=∫_0^πœ‹β–’γ€–π‘’^γ€–βˆ’cos〗⁑π‘₯ /(𝑒^γ€–βˆ’cos〗⁑π‘₯ + 𝑒^(γ€–βˆ’(βˆ’cos〗⁑π‘₯)) ) 𝑑π‘₯γ€— I=∫_0^πœ‹β–’γ€–π‘’^γ€–βˆ’cos〗⁑π‘₯ /(𝑒^γ€–βˆ’cos〗⁑π‘₯ + 𝑒^cos⁑π‘₯ ) 𝑑π‘₯γ€— Evaluate: ∫_0^πœ‹β–’γ€–π‘’^cos⁑π‘₯ /(𝑒^cos⁑π‘₯ + 𝑒^γ€–βˆ’cos〗⁑π‘₯ ) 𝑑π‘₯γ€— Let I=∫_0^πœ‹β–’γ€–π‘’^cos⁑π‘₯ /(𝑒^cos⁑π‘₯ + 𝑒^γ€–βˆ’cos〗⁑π‘₯ ) 𝑑π‘₯γ€— " " I= ∫_0^πœ‹β–’γ€–π‘’^cos⁑〖(πœ‹ βˆ’ π‘₯)γ€— /(𝑒^cos⁑〖(πœ‹ βˆ’ π‘₯)γ€— + 𝑒^γ€–βˆ’cos〗⁑〖(πœ‹ βˆ’ π‘₯)γ€— ) 𝑑π‘₯γ€— " " I=∫_0^πœ‹β–’γ€–π‘’^γ€–βˆ’cos〗⁑π‘₯ /(𝑒^γ€–βˆ’cos〗⁑π‘₯ + 𝑒^(γ€–βˆ’(βˆ’cos〗⁑π‘₯)) ) 𝑑π‘₯γ€— I=∫_0^πœ‹β–’γ€–π‘’^γ€–βˆ’cos〗⁑π‘₯ /(𝑒^γ€–βˆ’cos〗⁑π‘₯ + 𝑒^cos⁑π‘₯ ) 𝑑π‘₯γ€— Adding (1) and (2) i.e. (1) + (2) I+I=∫_0^πœ‹β–’γ€–π‘’^cos⁑π‘₯ /(𝑒^cos⁑π‘₯ + 𝑒^γ€–βˆ’cos〗⁑π‘₯ ) 𝑑π‘₯γ€— + ∫_0^πœ‹β–’γ€–π‘’^γ€–βˆ’cos〗⁑π‘₯ /(𝑒^γ€–βˆ’cos〗⁑π‘₯ + 𝑒^cos⁑π‘₯ ) 𝑑π‘₯γ€— 2I=∫_0^πœ‹β–’γ€–(𝑒^cos⁑π‘₯ + 𝑒^γ€–βˆ’cos〗⁑π‘₯ )/(𝑒^cos⁑π‘₯ + 𝑒^γ€–βˆ’cos〗⁑π‘₯ ) 𝑑π‘₯γ€— 2I =∫_0^πœ‹β–’γ€–1 .〗⁑𝑑π‘₯ 2I=[π‘₯]_0^πœ‹ 2I =πœ‹βˆ’0 2I =πœ‹ 𝐈=𝝅/𝟐

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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.