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
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Ex 7.3, 20 cos⁑2π‘₯/(cos⁑〖π‘₯ + sin⁑π‘₯ γ€— )^2 ∫1β–’cos⁑2π‘₯/(cos⁑π‘₯ + sin⁑π‘₯ )^2 =∫1β–’(cos^2⁑π‘₯ βˆ’ sin^2⁑π‘₯)/(cos⁑π‘₯ + sin⁑π‘₯ )^2 𝑑π‘₯ =∫1β–’(cos⁑π‘₯ βˆ’ sin⁑π‘₯ )(cos⁑π‘₯ + sin⁑π‘₯ )/(cos⁑π‘₯ + sin⁑π‘₯ )^2 𝑑π‘₯ =∫1β–’(cos⁑π‘₯ βˆ’ sin⁑π‘₯)/(cos⁑π‘₯ + sin⁑π‘₯ ) 𝑑π‘₯ Let cos⁑π‘₯+sin⁑π‘₯=𝑑 Differentiating w.r.t.x βˆ’sin⁑π‘₯+cos⁑π‘₯=𝑑𝑑/𝑑π‘₯ (cos⁑π‘₯βˆ’sin⁑π‘₯ )𝑑π‘₯=𝑑𝑑 𝑑π‘₯=1/((cos⁑π‘₯ βˆ’ sin⁑π‘₯ ) ) 𝑑𝑑 Thus, our equation becomes =∫1β–’((cos⁑π‘₯ βˆ’ sin⁑π‘₯ ))/𝑑 ×𝑑𝑑/(cos⁑π‘₯ βˆ’ sin⁑π‘₯ ) =∫1β–’1/𝑑 𝑑𝑑 =log⁑|𝑑|+𝐢 Putting value of 𝑑=π‘π‘œπ‘ β‘π‘₯+𝑠𝑖𝑛⁑π‘₯ =π’π’π’ˆβ‘|𝒄𝒐𝒔⁑𝒙+π’”π’Šπ’β‘π’™ |+π‘ͺ

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