Question $1$: $70$ Marks

Evaluate the following integrals.

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

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

$(1\mathrm{.}3){\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}si{n}^{5}xco{s}^{3}xdx$

$(6)$

$(1\mathrm{.}4){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}cos{h}^{-1}xdx$

$(5)$

$(1\mathrm{.}5){\displaystyle \underset{0}{\overset{1}{}}}\frac{1}{(4x^{2}9)!}dx,$ Using the substitution $x=\frac{3}{2}tanu$

$(8)$

$(1\mathrm{.}6){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}sin(lnx)dx$

$(6)$

$(1\mathrm{.}7){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{4x^{2}3x-2}{(x2)(x3)(x^{2}7x12)}dx$

$(1\mathrm{.}8){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}se{c}^4(\frac{1}{2}x)dx$

$(5)$

$(1\mathrm{.}9){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{-2x10}{5-4x-x^2}dx$

$(1\mathrm{.}10){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}cos5x\sqrt[\phantom{2}]{1sin5x}dx$

$(1\mathrm{.}11){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}co{t}^{3}xcs{c}^{5}xdx$

$2$

MAT$2691/101/0/2025$

$(1\mathrm{.}12){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x-1}{\sqrt[\phantom{2}]{x^2-4x3}}dx$

$(6)$