Example 40 - Chapter 7 Class 12 Integrals

Last updated at Dec. 16, 2024 by Teachoo

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Example 40 Evaluate ∫1▒(sin⁡〖2𝑥 cos⁡2𝑥 〗 𝑑𝑥)/(√(9 −cos^4⁡(2𝑥) ) ) 𝑑𝑥 ∫1▒(sin⁡〖2𝑥 cos⁡2𝑥 〗 𝑑𝑥)/(√(9 − cos^4⁡2𝑥 ) ) 𝑑𝑥 Let cos^2⁡2𝑥=𝑡 Differentiating both sides w.r.t.𝑥 –2.2 cos⁡2𝑥 sin⁡2𝑥=𝑑𝑡/𝑑𝑥 𝑑𝑥=𝑑𝑡/(–4 cos⁡2𝑥 sin⁡2𝑥 ) Hence, our equation becomes ∫1▒(sin⁡〖2𝑥 cos⁡2𝑥 〗 )/(√(9 − cos^4⁡2𝑥 ) ) 𝑑𝑥 =∫1▒〖sin⁡〖2𝑥 cos⁡2𝑥 〗/√(9 − 𝑡^2 ) 𝑑𝑥〗 =∫1▒〖(sin⁡2𝑥 cos⁡2𝑥)/√(9 − 𝑡^2 ) × 𝑑𝑡/(–4 cos⁡2𝑥 sin⁡2𝑥 )〗 =1/(−4) ∫1▒𝑑𝑡/√((3)^2 − (𝑡)^2 ) =(−1)/( 4) [sin^(−1)⁡〖𝑡/3+𝐶1〗 ] =(−1)/( 4) 𝑠𝑖𝑛^(−1) 𝑡/3−𝐶1/4 =(−𝟏)/( 𝟒) 𝒔𝒊𝒏^(−𝟏) [𝟏/𝟑 𝒄𝒐𝒔^𝟐 𝟐𝒙]+𝑪

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo

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