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Question
Evaluate : \[\int\limits_0^{2\pi} \cos^5 x dx\] .
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Solution
Let I =`int_0^(2x) cos ^5 x dx` ..... (1)
` cos^5( 2π - x )= cos^5 x `
It is known that,
`int_0^(2a) f (x) dx = 2 int _0^a f (x)dx, if f (2a - x ) = f(x)`
= 0 if f (2a - x = - f (x)
∴ `I = 2 int_0^π cos^5 x dx `
Now
\[f\left( \pi - x \right) = \cos^5 \left( \pi - x \right) = - \cos^5 x = - f\left( x \right)\]
⇒ I = 2(0 ) = 0 [ `cos^5(π - x) = - cos^5 x`]
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