Evaluate the following integral: `int_0^(pi//2)sqrt(cos x-cos^3x)"\ "(sec^2x-1)cos^2x\ dx`

Explanation

Let `I=int_0^(pi//2)sqrt(cos x(1-cos^2x))(sec^2x-1)cos^2x\ dx`

`I=int_0^(pi//2)sqrt(cos x(sin^2x))(tan^2x)cos^2x\ dx`

`I=int_0^(pi//2)sqrt(cos x)(sin^3x)dx`

`I=int_0^(pi//2)sqrt(cos x)(1-sin^2x)(sinx)dx`

Put `cosx=t,-sinxdx=dt`

When `x = 0, t = cos 0 = 1 `
When `x = π/2 , t = cos (π/2) = 0`
.
`I=int_1^0 sqrt(t)(1-t^2)(-dt)`

`I=int_1^0 -(t^(1/2)-t^(5/2)dt)`

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