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Ex 7.9, 5 Evaluate the integrals using substitution ∫_0^(πœ‹/2 )β–’sin⁑π‘₯/(1 + cos^2⁑π‘₯ )⁑〖 𝑑π‘₯γ€— ∫_0^(πœ‹/2 )β–’sin⁑π‘₯/(1 + cos^2⁑π‘₯ )⁑〖 𝑑π‘₯γ€— Put cos π‘₯=𝑑 Differentiating w.r.t.π‘₯ βˆ’sin⁑π‘₯=𝑑𝑑/𝑑π‘₯ 𝑑π‘₯=(βˆ’π‘‘π‘‘)/sin⁑π‘₯ Hence when π‘₯ varies from 0 to πœ‹/2, 𝑑 varies from 1 to 0 Therefore, we can write ∫_0^(πœ‹/2)β–’sin⁑π‘₯/(1+γ€– cos^2〗⁑π‘₯ ) 𝑑π‘₯=∫_1^0β–’γ€–sin⁑π‘₯/(1 + 𝑑^2 ) ((βˆ’π‘‘π‘‘)/sin⁑π‘₯ ) γ€— =βˆ’βˆ«_1^0▒𝑑𝑑/(1 + 𝑑^2 ) =βˆ’[tan^(βˆ’1)⁑𝑑 ]_1^0 =βˆ’[tan^(βˆ’1)⁑〖(0)βˆ’tan^(βˆ’1)⁑(1) γ€— ] =βˆ’[0βˆ’πœ‹/4] =βˆ’[βˆ’πœ‹/4] =𝝅/πŸ’

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.