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  1. Chapter 7 Class 12 Integrals
  2. Concept wise

Transcript

Misc 38 Prove that ∫_0^(πœ‹/4)β–’γ€–2 tanγ€—^3⁑π‘₯ 𝑑π‘₯=1βˆ’log⁑2 Solving L.H.S 2∫_0^(πœ‹/4)β–’γ€– tanγ€—^3⁑π‘₯ 𝑑π‘₯ = 2∫_0^(πœ‹/4)β–’γ€– tan π‘₯ tanγ€—^2⁑π‘₯ 𝑑π‘₯ = 2∫_0^(πœ‹/4)β–’γ€– tan π‘₯ (secγ€—^2⁑〖π‘₯βˆ’1)γ€— 𝑑π‘₯ = 2∫_0^(πœ‹/4)β–’γ€– tan π‘₯ secγ€—^2⁑π‘₯ 𝑑π‘₯βˆ’ 2∫_0^(πœ‹/4)β–’tan⁑〖π‘₯ 𝑑π‘₯γ€— 𝑰_𝟏 = 2∫_𝟎^(𝝅/πŸ’)β–’γ€– 𝐭𝐚𝐧 𝒙 𝒔𝒆𝒄〗^πŸβ‘π’™ 𝒅𝒙 Let t = tan x 𝑑𝑑/𝑑π‘₯ = 〖𝑠𝑒𝑐〗^2 x dt = 〖𝑠𝑒𝑐〗^2x dx Substituting, 2∫1_0^1▒〖𝑑 𝑑𝑑〗 = 2 [𝑑^2/2]_0^1 = 2 (1/2βˆ’0) =1 𝑰_𝟐= 2∫_𝟎^(𝝅/πŸ’)▒𝒕𝒂𝒏⁑〖𝒙 𝒅𝒙〗 2∫_𝟎^(𝝅/πŸ’)▒𝒕𝒂𝒏⁑〖𝒙 𝒅𝒙〗 = 2 ["log" |saec⁑π‘₯ |]_0^(πœ‹/4) = 2 ("log" √2βˆ’0) = 2 (log 2^(1/2)) = 2 (1/2 "log " 2) = log 2 Hence, = 𝐼_1βˆ’πΌ_2 = 1 βˆ’ log 2 = R.H.S Hence, proved.

Chapter 7 Class 12 Integrals
Concept wise

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.