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Question
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`
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Solution
Let I = `int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`
I = `int_0^(pi/2) cos^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`
`2I = int_0^(pi/2) (sin^(3/2)x/(sin^(3/2)x+cos^(3/2) x)+cos^(3/2)x/(sin^(3/2)x + cos^(3/2)x)) dx`
Simplify the numerator:
`(sin^(3/2)x+cos^(3/2) x)/(sin^(3/2)x+cos^(3/2)) = 1`
`2I = int_0^(pi/2) 1 dx`
`int_0^(pi/2) 1 dx = [x]_0^(pi/2)=pi/2 - 0 = pi/2`
`2I = pi/2`
`I=pi/4`
`pi/4`
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