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Ex 7.10, 5 - Ex 7.10, 5
Evaluate the integrals using substitution
∫_0^(πœ‹/2  )β–’sin⁑π‘₯/(1 + cos^2⁑π‘₯ )⁑〖 𝑑π‘₯γ€—   - Definate Integration - By Substitution

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  1. Chapter 7 Class 12 Integrals
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Ex 7.10, 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 cosπ‘₯=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 then 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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