Question $1$: $34$ Marks

Determine the following integrals:

$(1\mathrm{.}1){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{10}{25-x^2}dx.$

$(1\mathrm{.}2){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}ln(e^{2x^2-5})dx$

$(1\mathrm{.}3){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{9}{(1-3x{)}^4}dx.$

$(1\mathrm{.}4){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{-3dx}{(2+x)(1+2x)}.$

$(1\mathrm{.}5){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^3+4x^2}{2x}dx.$

$(1\mathrm{.}6),{\displaystyle \underset{\frac{}{2}}{\overset{}{}}}sin3xdx.$

$(1\mathrm{.}7){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{3x}(e^{3x-4}{)}^{4}dx.$

$(1\mathrm{.}8),{\displaystyle \underset{\frac{}{4}}{\overset{}{}}}4cos2xdx$

Question $2$: $7$ Marks

$An$ object moves with variable acceleration, $a,$ given $by$ the formula $2t^2-3t-5.$

$(2\mathrm{.}1)$ Find $an$ expression for the velocity $of$ the object.

$(2\mathrm{.}2)$ Find $an$ expression for the displacement $of$ the object

$32$

Question $3$: $6$ Marks

Find the area bounded $by$ the graph $y=x^2-x-6,$ the $x-$axis and the lines $x=-1$ and $x=3.$