Ex 7.3, 20 - Chapter 7 Class 12 Integrals

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Ex 7.3, 20 Integrate the function 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 (𝑐𝑜𝑠⁡2𝜃=〖𝑐𝑜𝑠〗^2⁡𝜃−〖𝑠𝑖𝑛〗^2⁡𝜃) −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