Evaluate the following integrals: `int_0^((pi)^(2//3))sqrt(x)cos^2x^(3//2)dx`

Explanation

Given

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

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

Let `x^(3/2)=t implies (3/2)x^(1/2)dx=dt implies sqrtxdx=(2/3)dt`

And limit changes from `(0,(pi)^(2/3))` to `(0,pi)`

`I=int_0^(pi)(2/3)cos^2tdt`

`I=(1/3)int_0^(pi)(1+cos2t)dt`

`I=(1/3)[t+(sin2t)/2]_0^(pi)`

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