i$)$ Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\sqrt[\phantom{2}]{(x^2-a^2)}dx$

$(5$ marks$)$

ii$)$ Evaluate ${\displaystyle \underset{1}{\overset{2}{}}}\frac{2}{x^2\sqrt[\phantom{2}]{(1+x^2)}}dx,$ correct to $3$ significant figures

$(5$ marks$)$

iii$)$ Given that $y=2x{e}^{-3x}$ show that $\frac{d^{2}y}{d{x}^2}+6\frac{dy}{dx}+9y=0.$

$(5$ marks$)$

iv$)$ Determine ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{dx}{cosx}$

$(5$ marks$)$

v$)$ When determining the surface tension of a liquid, the radius of curvature, $,$ of part of the surface is given by $=\frac{\sqrt[\phantom{2}]{[1+(\frac{dy}{dx}{)}^2{]}^3}}{\frac{d^{2}y}{d{x}^2}}.$ Find the radius of curvature of the part of the surface having the parametric equations $x=3t^2$ and $y=6t$ at the point $t=2.$

vi$)$ Use trapezoidal rule to evaluate ${\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}\frac{1}{1+sinx}dx$ using $6$ intervals

$(5$ marks$)$

$(5$ marks$)$