Ex 7.3, 20 - Integrate cos 2x / (cos x + sin x)2  - Ex 7.3

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

Transcript

Ex 7.3, 20 cos⁡2𝑥/(cos⁡〖𝑥 + sin⁡𝑥 〗 )^2 ∫1▒cos⁡2𝑥/(cos⁡𝑥 + sin⁡𝑥 )^2 =∫1▒(cos^2⁡𝑥 − sin^2⁡𝑥)/(cos⁡𝑥 + sin⁡𝑥 )^2 𝑑𝑥 =∫1▒(cos⁡𝑥 − sin⁡𝑥 )(cos⁡𝑥 + sin⁡𝑥 )/(cos⁡𝑥 + sin⁡𝑥 )^2 𝑑𝑥 =∫1▒(cos⁡𝑥 − sin⁡𝑥)/(cos⁡𝑥 + sin⁡𝑥 ) 𝑑𝑥 Let cos⁡𝑥+sin⁡𝑥=𝑡 Differentiating w.r.t.x −sin⁡𝑥+cos⁡𝑥=𝑑𝑡/𝑑𝑥 (cos⁡𝑥−sin⁡𝑥 )𝑑𝑥=𝑑𝑡 𝑑𝑥=1/((cos⁡𝑥 − sin⁡𝑥 ) ) 𝑑𝑡 Thus, our equation becomes =∫1▒((cos⁡𝑥 − sin⁡𝑥 ))/𝑡 ×𝑑𝑡/(cos⁡𝑥 − sin⁡𝑥 ) =∫1▒1/𝑡 𝑑𝑡 =log⁡|𝑡|+𝐶 Putting value of 𝑡=𝑐𝑜𝑠⁡𝑥+𝑠𝑖𝑛⁡𝑥 =𝒍𝒐𝒈⁡|𝒄𝒐𝒔⁡𝒙+𝒔𝒊𝒏⁡𝒙 |+𝑪

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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.