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Question
Evaluate the following integrals using properties of integration:
`int_(- pi/4)^(pi/4) sin^2x "d"x`
Solution
I = `int_(- pi/4)^(pi/4) sin^2x "d"x`
f(x) = sin2x
f(– x) = sin2(– x) = sin2x
f(x) = f(– x)
f(x) is an even function
`int_(- pi/4)^(pi/4) sin^2x "d"x = 2 int_0^(pi/4) sin^2 x "dx`
`int_(- "a")^"a" f(x) "d"x = 2int_0^"a"f(x) "d"x`
= `2 int_0^(pi/4) [(1 - cos 2x)/2] "d"x`
= `2/2 [x - (sin 2x)/2]_0^(pi/4)`
= `pi/4 - 1/2`
I = `(pi - 2)/4`
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