Ex 7.3, 14 - Integrate cos x - sin x / 1 + sin 2x - Ex 7.3

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.3, 14 cos⁑〖π‘₯ βˆ’ sin⁑π‘₯ γ€—/(1 + sin⁑2π‘₯ ) ∫1β–’cos⁑〖π‘₯ βˆ’ sin⁑π‘₯ γ€—/(1 + sin⁑2π‘₯ ) 𝑑π‘₯ =∫1β–’cos⁑〖π‘₯ βˆ’γ€– sin〗⁑π‘₯ γ€—/(1 + 2 sin⁑π‘₯ cos⁑π‘₯ ) 𝑑π‘₯ =∫1β–’cos⁑〖π‘₯ βˆ’γ€– sin〗⁑π‘₯ γ€—/(sin^2⁑π‘₯ + cos^2⁑π‘₯ + 2 sin⁑cos⁑π‘₯ ) 𝑑π‘₯ =∫1β–’cos⁑〖π‘₯ βˆ’γ€– sin〗⁑π‘₯ γ€—/(sin⁑π‘₯ + cos⁑π‘₯ )^2 𝑑π‘₯ Let sin⁑π‘₯+cos⁑π‘₯=𝑑 Differentiating w.r.t.x 𝑑(sin⁑π‘₯ + cos⁑π‘₯ )/𝑑π‘₯=𝑑𝑑/𝑑π‘₯ cos π‘₯βˆ’sin⁑π‘₯=𝑑𝑑/𝑑π‘₯ 𝑑π‘₯=𝑑𝑑/(cos π‘₯ βˆ’ sin⁑π‘₯ ) Thus, our equation becomes ∫1β–’((cos⁑〖π‘₯ βˆ’ sin⁑π‘₯ γ€— ))/(sin⁑π‘₯ + cos⁑π‘₯ )^2 𝑑π‘₯ =∫1β–’((cos⁑〖π‘₯ βˆ’ sin⁑π‘₯ γ€— ))/𝑑^2 ×𝑑𝑑/cos⁑〖π‘₯ βˆ’ sin⁑π‘₯ γ€— =∫1▒𝑑𝑑/𝑑^2 =∫1▒𝑑^(βˆ’2) 𝑑𝑑 =𝑑^(βˆ’2 +1)/(βˆ’2 + 1) +𝐢 =𝑑^(βˆ’1)/(βˆ’1) +𝐢 =(βˆ’1)/𝑑 +𝐢 Putting value of 𝑑=𝑠𝑖𝑛⁑π‘₯+π‘π‘œπ‘  π‘₯ =(βˆ’πŸ)/(𝐬𝐒𝐧⁑𝒙 + 𝒄𝒐𝒔⁑𝒙 ) +π‘ͺ

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