I need solutions ...
$\displaystyle \int_0^2 \frac{x}{(x-1)^3}dx$
$\displaystyle \int_0^2 \frac{1}{\sqrt{x(2-x)}}dx$
1. Expand the denominator and then divide each term in the denominator by $\displaystyle x$ to simplify. Else use Integration by Parts with $\displaystyle u = x$ and $\displaystyle dv = (x - 1)^{-3}$.
2. Expand the denominator, Complete the Square and use a Trigonometric Substitution.
You still have not shown any work. Try to follow some of the hints you were given.
For the first one, I'd use substitution, $\displaystyle u = x - 1$
For the second, substitution $\displaystyle u = \sqrt x$. you will get it in a form where you can use $\displaystyle \int \frac {1}{\sqrt{a^2 - x^2}}~dx = \sin^{-1} \frac xa + C$ to finish off the problem