Integration Full Chapter Explained - Integration Class 12 - Everything you need

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise

Transcript

Ex 7.11, 11 By using the properties of definite integrals, evaluate the integrals : ∫_((βˆ’ πœ‹)/2)^(πœ‹/2)β–’γ€– sin^2〗⁑π‘₯ 𝑑π‘₯ This is of form ∫_(βˆ’π‘Ž)^π‘Žβ–’π‘“(π‘₯)𝑑π‘₯ where 𝑓(π‘₯)=sin^2⁑π‘₯ 𝑓(βˆ’π‘₯)=sin^2⁑(βˆ’π‘₯)=(βˆ’π‘ π‘–π‘›π‘₯)^2=sin^2⁑π‘₯ ∴ 𝑓(π‘₯)=𝑓(βˆ’π‘₯) Using the Property ∫_(βˆ’π‘Ž)^π‘Žβ–’γ€–π‘“(π‘₯)𝑑π‘₯=2,γ€— ∫_0^π‘Žβ–’γ€–π‘“(π‘₯)𝑑π‘₯ γ€— if f(βˆ’π‘₯)=𝑓(π‘₯) ∴ ∫_((βˆ’πœ‹)/2)^(πœ‹/2)β–’γ€–sin^2⁑〖π‘₯ 𝑑π‘₯γ€—=2∫_0^(πœ‹/2)β–’γ€–γ€–π’”π’Šπ’γ€—^𝟐 𝒙 𝑑π‘₯γ€—γ€— =2∫_0^(πœ‹/2)β–’[(𝟏 βˆ’ π’„π’π’”β‘πŸπ’™)/𝟐]𝑑π‘₯ =∫_0^(πœ‹/2)β–’γ€–(1βˆ’cos⁑2π‘₯ ) 𝑑π‘₯γ€— = [π‘₯ βˆ’sin⁑2π‘₯/2]_0^(πœ‹/2) = [πœ‹/2βˆ’sin⁑2(πœ‹/2)/2]βˆ’ [0βˆ’sin⁑〖2(0)γ€—/2] = πœ‹/2βˆ’sinβ‘πœ‹/2βˆ’0 = πœ‹/2βˆ’0+0 = 𝝅/𝟐 ∡ cos 2x = 1 βˆ’ 2 〖𝑠𝑖𝑛〗^2 π‘₯ β‡’ 2 〖𝑠𝑖𝑛〗^2 π‘₯ = 1 βˆ’ cos 2x β‡’ 〖𝑠𝑖𝑛〗^2 π‘₯ = "1 βˆ’ cos 2x" /2

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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.