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Question
By using the properties of the definite integral, evaluate the integral:
`int_(pi/2)^(pi/2) sin^7 x dx`
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Solution
Let f (x) = sin7 x.
sin x is an odd function
i.e. if h (x) = sin x
⇒ h (-x) = sin (-x)
= - sin (x) = -h (x)
⇒ odd power of sin x is odd
⇒ f (x) is an odd function of x.
⇒ `int_(-pi/2)^(pi/2) sin^7 x dx = 0` .... [∵ If f (x) is odd ⇒`int_-a^a` f (x) dx = 0]
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