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Question
Evaluate the following using properties of definite integral:
`int_(- pi/2)^(pi/2) sin^2theta "d"theta`
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Solution
Let f(θ)= sin2θ
f(– θ) = sin2(– θ)
= [sin (– θ)]2
= [– sin θ]2
= sin2θ
f(– θ) = f(θ)
∴ f(θ) is an even function
`int_(- pi/2)^(pi/2) sin^2theta "d"theta = 2 xx int_0^(pi/2) sin^2theta "d"theta`
= `2 xx int_0^(pi/2) ((1 - cos 2theta)/2) "d"theta`
= `2 xx 1/2 int_0^(pi/2) (1 - cos 2theta) "d"theta`
= `[theta - (sin 2theta)/2]_0^(pi/2)`
= `[pi/2 - (sin2(pi/2))/2] - [0]`
= `pi/2 - 0`
= `pi/2`
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