The value of `int_(0)^(pi/2) cos ^5 x dx` has to be determined.
`int_(0)^(pi/2) cos ^5 x dx`
=> `int_(0)^(pi/2) cos^4 x*cos x dx`
=> `int_(0)^(pi/2) (cos^2 x)^2 *cos x dx`
=> `int_(0)^(pi/2) (1 - sin^2x)^2 *cos x dx`
=> `int_(0)^(pi/2) (1 - 2*sin^2x + sin^4x) *cos x dx`
Let `sin...
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The value of `int_(0)^(pi/2) cos ^5 x dx` has to be determined.
`int_(0)^(pi/2) cos ^5 x dx`
=> `int_(0)^(pi/2) cos^4 x*cos x dx`
=> `int_(0)^(pi/2) (cos^2 x)^2 *cos x dx`
=> `int_(0)^(pi/2) (1 - sin^2x)^2 *cos x dx`
=> `int_(0)^(pi/2) (1 - 2*sin^2x + sin^4x) *cos x dx`
Let `sin x = y`
`dy = cos x* dx`
`sin 0 = 0` and `sin(pi/2) = 1`
The required integral is changed to:
`int_(0)^(1) 1 - 2*y^2 + y^4 dy`
=> `y - 2y^3/3 + y^5/5` between y = 0 and y = 1
=> `1 - 2/3 + 1/5`
=> `8/15`
The value of `int_(0)^(pi/2) cos ^5 x dx` = `8/15`