Evaluate : 2 π ∫ 0 Cos 5 X D X .

Question

Evaluate : \[\int\limits_0^{2\pi} \cos^5 x dx\] .

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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Definite Integrals

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2016-2017 (March) Foreign Set 3

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Evaluate : 2 π ∫ 0 Cos 5 X D X .

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