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Chapter 7 Class 12 Integrals
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Ex 7.10, 4 By using the properties of definite integrals, evaluate the integrals : ∫_0^(πœ‹/2)β–’(cos^5⁑π‘₯ 𝑑π‘₯)/(sin^5⁑π‘₯ + cos^5⁑π‘₯ ) Let I=∫_0^(πœ‹/2)β–’γ€–cos^5⁑π‘₯/(sin^5⁑π‘₯ + cos^5⁑π‘₯ ) 𝑑π‘₯γ€— I= ∫_0^(πœ‹/2)β–’γ€–(cos^5 (πœ‹/2 βˆ’ π‘₯))/(〖𝑠𝑖𝑛〗^5 (πœ‹/2 βˆ’ π‘₯) + γ€–π‘π‘œπ‘ γ€—^5 (πœ‹/2 βˆ’ π‘₯) ) 𝑑π‘₯γ€— ∴ I = ∫_0^(πœ‹/2)β–’γ€– sin^5⁑π‘₯/(cos^5⁑π‘₯ + sin^5⁑π‘₯ ) 𝑑π‘₯γ€— Adding (1) and (2) i.e. (1) + (2) I+I=(γ€–π‘π‘œπ‘ γ€—^5 π‘₯)/(〖𝑠𝑖𝑛〗^5 π‘₯ + γ€–π‘π‘œπ‘ γ€—^5 π‘₯) 𝑑π‘₯+∫_0^(πœ‹/2)β–’γ€–sin^5⁑π‘₯/(cos^5⁑π‘₯ + sin^5⁑π‘₯ ) 𝑑π‘₯γ€— 2I=∫_0^(πœ‹/2)β–’γ€–[(γ€–π‘π‘œπ‘ γ€—^5 π‘₯ + 〖𝑠𝑖𝑛〗^5 π‘₯)/(γ€–π‘π‘œπ‘ γ€—^5 π‘₯ + 〖𝑠𝑖𝑛〗^5 π‘₯)] 𝑑π‘₯γ€— 2I= ∫_0^(πœ‹/2)β–’γ€– 𝑑π‘₯γ€— I=1/2 ∫_0^(πœ‹/2)β–’γ€– 𝑑π‘₯γ€— I=1/2 [π‘₯]_0^(πœ‹/2) I=1/2 [πœ‹/2βˆ’0] 𝑰=𝝅/πŸ’

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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.