$(4)$ Intenrare ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}ln(x\sqrt[\phantom{2}]{x^2-1})dx$ using intermation by parts Hint entegrare directily, or first usetce identify $cos{h}^{-1}x=ch(x\sqrt[\phantom{2}]{x^2-1}),$ usere $x>1,$ whethat $\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}(cos{h}^{-1}x)=\frac{1}{\sqrt[\phantom{2}]{x^2-1}}$