Evaluate the following integral: `int_0^(pi//4)cos^4xsin^3x\ dx`

Explanation

Given:`int_0^(pi/4)cos^4xsin^3x dx`

`I=int_0^(pi/4)cos^4xsin^3xdx`

`I=int_0^(pi/4)cos^4xsin^2x sinx dx`

`I=int_0^(pi/4)cos^4x(1-cos^2x)sinx dx`

Let `cosx=t implies -sinx dx=dt`

Then

`I=-int_1^(1/(sqrt2))t^4(1-t^2)dt`

`I=int_1^(1/(sqrt2))t^4(t^2-1)dt`

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