Evaluate:
(i) `intsqrt(tanx)(1+tan^2x)\ dx`
(ii) `int{f\ (a x+b)}^nf^(prime)(a x+b)dx ,\ \ n!=-1`

Explanation

`(i)`We have,
`I=intsqrt(tanx)(1+tan^2x)\ dx`
`I=intsqrt(tanx)sec^2x dx`
Put `tanx=t`
`sec^2x dx=dt`
Thus, `I =int sqrt(t) dt`
`I=t^(3/2)/(3/2)+c`
`I=2/3 (tanx)^(3/2) +c`


`(ii)`Given
`int{f\ (a x+b)}^nf^(prime)(a x+b)dx ,\ \ n!=-1`
`f(ax+b)->t`
`af'(ax+b)->(dt)/(dx)`
`f'(ax+b)dx->(dt)/a`
therefore,
`int(t^n*dt)/a`
`=(t^(n+1))/(a(n+1))+c`
`=((f(ax+b))^(n+1))/(a(n+1))+c`