Evaluate : (i) `int(logx)/(x)dx` (ii) `int(sec^(2)(logx))/(x)dx`
(iii) `(e^(tan^(-1)x))/((1+x^(2)))dx` (iv) `int(sinsqrt(x))/(sqrt(x))dx`

Explanation

(i) Put log x=t so that `(1)/(x)dx=dt`.
`:.int(logx)/(x)dx=inttdt=(1)/(2)t^(2)+C=(1)/(2)(logx)^(2)+C`.
(ii) Put log x=t so that `(1)/(x)dx=xt`.
`:.int(sec^(2)(logx))/(x)dx=intdt=tant+C=tant+C=tan(logx)+C`.
(iii) Put `tan^(-1)x=t` so that `(1)/((1+x^(2))dx=dt`.
`:.int(e^(tan^(-1)x))/((1+x^(2)))dx=inte^(t)dt=e^(t)+C=e^(tan^(-1)x)+C`.
(iv) Put `sqrt(x)=t` so that `(1)/(2)x^(-1//2)dx=dt`
`or(1)/(sqrt(x))dx=2dt`.
`:.int(sinsqrt(x))/(sqrt(x))dx=2intsintdt=2(-cost)+C=-2cost+C=-2cossqrt(x)+C`.