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Question
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (cos^5 xdx)/(sin^5 x + cos^5 x)`
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Solution
Let `I = int_0^(pi/2) (cos^5 x)/ (sin^5 x + cos ^5 x) dx` ....(i)
Also, `I = int_0^(pi/2) (cos^5 (pi/2 - x))/(sin^5 (pi/2 - x) + cos^5 (pi/2 - x)) dx`
`[∵ int_0^a f (x) dx = int_0^a f (a - x) dx]`
`= int_0^(pi/2) (sin^5 x)/ (cos^5x + sin^5 x) dx` ....(ii)
Adding (i) and (ii), we have
`2 I = int_0^(pi/2) (cos^5x)/(cos^5x + sin^5 x) dx + int_0^(pi/2) (sin^5x)/ (cos^5 x + sin^5 x) dx`
`= int_0^(pi/2) (cos^5 x + sin^5 x)/ (cos^5 x + sin^5 x) dx`
`= int_0^(pi/2) 1 dx = [x]_0^(pi/2) = pi/2`
Hence, `I = pi/4`
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