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Question
By using the properties of the definite integral, evaluate the integral:
`int_((-pi)/2)^(pi/2) sin^2 x dx`
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Solution
Let`I = int_(-pi//2)^(pi//2) sin^2 x dx`
`= 2 int_0^(pi//2) sin^2 x dx` ...(i) ...(∵ sin2 x is a function)
Then `I = 2 int_0^(pi//2) sin^2 (pi/2 - x) dx`
`= int_0^(pi//2) cos^2 x dx` ...(ii) `[because int_0^a f(x) = int_0^a f(a - x) dx]`
On adding equations (i) and (ii)
`2I = 2 int_0^(pi//2) (sin^2 x + cos^2 x) dx`
`2I = 2 int_0^(pi//2) 1 dx`
`=> 2I = 2 [x]_0^(pi//2)`
`=> 2I = 2 xx pi/2`
Hence, `I = pi/2`
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