Evaluate the following integral: `int_0^1cos^(-1)x\ dx`

Explanation

Given:`int_0^1cos^(-1)x\ dx`

`I=int_0^1cos^(-1)x\ dx`

`I=int_0^1 1.cos^(-1)x\ dx`

`I=(cos^(-1)x [x])_0^1-int_0^1 -x/(sqrt(1-x^2))dx`

`I=[xcos^(-1)x]_0^1-int_0^1 -x/(sqrt(1-x^2))dx`

Now let `1-x^2=t implies -2xdx=dt implies -xdx=(dt)/2`

When `x=0, t=1`, and when `x=1,t=0`

`I=[1.cos^(-1) (1)-0]-int_1^0 (dt)/(2sqrtt)`

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