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

Ex 7.3, 20 - Chapter 7 Class 12 Integrals - Part 2

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Ex 7.3, 20 Integrate the function cos⁑2π‘₯/(cos⁑〖π‘₯ γ€—+ sin⁑π‘₯ )^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 (π‘π‘œπ‘ β‘2πœƒ=γ€–π‘π‘œπ‘ γ€—^2β‘πœƒβˆ’γ€–π‘ π‘–π‘›γ€—^2β‘πœƒ) 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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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.