Ex 7.10, 4 By using the properties of definite integrals, evaluate the integrals : ∫_0^(𝜋/2)▒(cos^5⁡𝑥 𝑑𝑥)/(sin^5⁡𝑥 + cos^5⁡𝑥 ) Let I=∫_0^(𝜋/2)▒〖cos^5⁡𝑥/(sin^5⁡𝑥 + cos^5⁡𝑥 ) 𝑑𝑥〗 I= ∫_0^(𝜋/2)▒〖(cos^5 (𝜋/2 − 𝑥))/(〖𝑠𝑖𝑛〗^5 (𝜋/2 − 𝑥) + 〖𝑐𝑜𝑠〗^5 (𝜋/2 − 𝑥) ) 𝑑𝑥〗 ∴ I = ∫_0^(𝜋/2)▒〖 sin^5⁡𝑥/(cos^5⁡𝑥 + sin^5⁡𝑥 ) 𝑑𝑥〗 Adding (1) and (2) i.e. (1) + (2) I+I=(〖𝑐𝑜𝑠〗^5 𝑥)/(〖𝑠𝑖𝑛〗^5 𝑥 + 〖𝑐𝑜𝑠〗^5 𝑥) 𝑑𝑥+∫_0^(𝜋/2)▒〖sin^5⁡𝑥/(cos^5⁡𝑥 + sin^5⁡𝑥 ) 𝑑𝑥〗 2I=∫_0^(𝜋/2)▒〖[(〖𝑐𝑜𝑠〗^5 𝑥 + 〖𝑠𝑖𝑛〗^5 𝑥)/(〖𝑐𝑜𝑠〗^5 𝑥 + 〖𝑠𝑖𝑛〗^5 𝑥)] 𝑑𝑥〗 2I= ∫_0^(𝜋/2)▒〖 𝑑𝑥〗 I=1/2 ∫_0^(𝜋/2)▒〖 𝑑𝑥〗 I=1/2 [𝑥]_0^(𝜋/2) I=1/2 [𝜋/2−0] 𝑰=𝝅/𝟒

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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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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