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

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  1. Chapter 7 Class 12 Integrals
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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‘₯ 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‹/2.2 ∴ 𝐈 = 𝝅/πŸ’

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.