`int(x^2)/(sqrt(1-x))\ dx` (ii) `intx(1-x)^(23)\ dx`

Explanation

(i)
`Let I=int frac{x^{2}}{sqrt{x-1}} dx`

Substituting `x-1=t` and `d x=d t`, we get,

`I=int frac{(t+1)^{2}}{sqrt{t}} dx`

`=int frac{t^{2}+1+2 t}{sqrt{t}} d t`

`=int(t^{frac{3}{2}}+t^{-frac{1}{2}}+2 t^{-frac{1}{2}}) d t`

`=frac{2}{5} t^{frac{5}{2}}+2 t^{frac{1}{2}}+frac{4}{3} t^{frac{3}{2}}+c`

`=frac{6 t^{frac{5}{2}}+30 t^{frac{1}{2}}+20 t^{frac{3}{2}}}{15}+c`

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