Evaluate: (i) `int1/(1+sqrt(x))\ dx` (ii) `inte^cos^(2x)sin2x\ dx`

Explanation

`(i) int1/(1+sqrt(x))\ dx`
taking `1+sqrt(x)=t`
`=>sqrtx =t-1`
`=>x=(t-1)^2`
`=>dx=2(t-1)dt`
so, `int (2(t-1))/t dt`
`2 int 1dt -2 int 1/t dt`
`=2t-2 log|t|+c`
`=2(1+sqrtx)-2 log|1+sqrtx|+c`

`(ii) inte^(cos^2x)sin2x\ dx`
`int e^(cos^2x) sin 2x dx`
taking `t= e^(cos^2x)`
`=>e^(cos^2x)(2cosx)(-sinx)dx=dt`
`=>-e^(cos^2x)(sin2x)dx=dt`
so,`int e^(cos^2x)(2sinx cosx)dx`
`int -dt`
`=-t+c`
`=-e^(cos^2x)+c`