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Example 34 - Evaluate integral sin4 x / sin4 x + cos4 x dx

Example 34 - Chapter 7 Class 12 Integrals - Part 2


Transcript

Example 34 Evaluate ∫_0^(πœ‹/2)β–’sin^4⁑π‘₯/(sin^4⁑π‘₯ + cos^4⁑π‘₯ ) 𝑑π‘₯ Let I =∫_0^((πœ‹ )/2)β–’γ€–(〖𝑠𝑖𝑛〗^4 π‘₯)/〖〖𝑠𝑖𝑛〗^4 π‘₯〗⁑〖+ γ€–π‘π‘œπ‘ γ€—^4 π‘₯γ€— 𝑑π‘₯γ€— ∴ I =∫_0^((πœ‹ )/2)β–’sin^4⁑(πœ‹/2 βˆ’ π‘₯)/(〖〖𝑠𝑖𝑛〗^4 π‘₯〗⁑〖 (πœ‹/2 βˆ’ π‘₯) γ€—+ γ€–γ€–π‘π‘œπ‘ γ€—^4 π‘₯〗⁑(πœ‹/2 βˆ’ π‘₯) ) 𝑑π‘₯ I = ∫_0^((πœ‹ )/2)β–’(γ€–π‘π‘œπ‘ γ€—^4 π‘₯)/γ€–γ€–π‘π‘œπ‘ γ€—^4 π‘₯〗⁑〖+〖𝑠𝑖𝑛〗^4 π‘₯γ€— 𝑑π‘₯ Using the property P4 ∫_0^π‘Žβ–’γ€–π‘“(π‘₯)𝑑π‘₯=γ€— ∫_0^π‘Žβ–’π‘“(π‘Žβˆ’π‘₯)𝑑π‘₯ Using :- sin (πœ‹/2βˆ’πœƒ)=cosβ‘πœƒ & cos (πœ‹/2βˆ’πœƒ)=sinβ‘πœƒ Adding (1) and (2) i.e. (1) + (2) I + I = ∫_0^((πœ‹ )/2)β–’(〖𝑠𝑖𝑛〗^4 π‘₯)/(〖𝑠𝑖𝑛〗^4 π‘₯ +γ€–π‘π‘œπ‘ γ€—^4 π‘₯) 𝑑π‘₯+∫_0^(πœ‹/2)β–’γ€–(π‘π‘œπ‘  π‘₯)/(〖𝑠𝑖𝑛〗^4 π‘₯ +γ€–π‘π‘œπ‘ γ€—^4 π‘₯).γ€— 𝑑π‘₯ 2I = ∫_0^((πœ‹ )/2)β–’γ€–(〖𝑠𝑖𝑛〗^4 π‘₯ + γ€–π‘π‘œπ‘ γ€—^4 π‘₯)/(〖𝑠𝑖𝑛〗^4 π‘₯ +γ€–π‘π‘œπ‘ γ€—^4 π‘₯).γ€— 𝑑π‘₯ 2I = ∫_0^((πœ‹ )/2)▒𝑑π‘₯" " 2I = [π‘₯]_0^(πœ‹/2) 2I = [πœ‹/2βˆ’0] I = πœ‹/(2 Γ— 2) ∴ 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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.