QUESTION $1(5$ Marks$)$

Use integration $by$ partial fraction $to$ solve

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

QUESTION $2(8$ Marks$)$

Find the values $of$ the constants A and $Bif$

${\displaystyle \underset{3}{\overset{9}{}}}cos{h}^{-1}(\frac{x}{3})dx=$Aln$(3+\sqrt[\phantom{2}]{8})-B\sqrt[\phantom{2}]{2}$

QUESTION $3(5$ Marks$)$

Evaluate the following integral

${\displaystyle \underset{1}{\overset{}{}}}\frac{e^x}{\sqrt[\phantom{2}]{e^{2x}-1}}dx$

$(5$ marks$)$

QUESTION $4(12$ Marks$)$

$a{\displaystyle \underset{n=1}{\overset{}{}}}\frac{n{e}^n}{5^{n+1}}.$

$(5$ marks$)$

$bf(x)=(1+x{)}^{\frac{3}{2}}.$

$i.$ Show that the first four terms $of$ the Maclaurin series for the above

function $is$

$(1+x{)}^{\frac{3}{2}}=1+\frac{3}{2}x+\frac{3}{8}{x}^2-\frac{1}{16}{x}^3+$cdots

$ii.$ Hence, estimate ${\displaystyle \underset{0}{\overset{0\mathrm{.}5}{}}}(1-x^2{)}^{\frac{3}{2}}dx.$