Question

Evaluate `int tan^(-1) sqrtx dx`

Answer 1

`I = int tan^(-1) sqrtx dx`

Put `sqrtx = t`

`1/(2sqrtx) dx = dt`

`dx = 2sqrtx dt -> dx = 2t dt`

`I = int 2t tan^(-1) t dt`

`I = 2[t^2/2 tan^(-1) t - 1/2 int t^2/(1+ t^2) dt]`

`I = 2[t^2/2 tan^1 t - 1/2 int [(1+t^2)/(1+t^2) - 1/(1+t^2)]dt]`

`I = [t^2 tan^(-1) t - t + tan^(-1) t] + c`

`I = t^2 tan^(-1) t  - t + tan^(-1) t + c`

`I =(x + 1)tan^-1 sqrtx - sqrtx +c `