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By using the properties of definite int...

By using the properties of definite integrals, evaluate the integrals`int_0^(pi/2)(cos^5x dx)/(sin^5+cos^5x)`

Text Solution

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Let` I=int_0^(pi/2)cos^(5)x/(​​sin^(5)x+cos^(5)x)dx ............(1)`
⇒`I=int_0^(pi/2)cos^(5)(pi/2-x)/(​​sin^(5)(pi/2-x)+cos^(5)(pi/2-x))dx `
⇒`I=int_0^(pi/2)sin^(5)x/(​​sin^(5)x+cos^(5))dx .......... (2)x`
Adding (1) and (2), we obtain
`2I=int_0^(pi/2)(sin^(5)x+cos^(5)x)/(sin^(5)x+cos^(5)x)`
⇒`2I=int_0^(pi/2)1⋅dx`
⇒`2I=π/2`
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