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