ASSIGNMENT $02$

Due date: Friday, $15$ August $2025$

Total Marks: $80$

ONLY FOR YEAR MODULE

This assignment covers work from MAT$2691-$Module $2$ in study guide $2$

Question $1$: $70$ Marks

Evaluate the following integrals.

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

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

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

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

$(1\mathrm{.}5){\displaystyle \underset{0}{\overset{\frac{3}{2}}{}}}\frac{1}{(4x^2+9{)}^{\frac{3}{2}}}dx$

Using the substitution $x=\frac{3}{2}tanu$

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

$(1\mathrm{.}7){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{4x^2+3x-2}{(x+2)(x+3)(x^2+7x+12)}dx$

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

$(1\mathrm{.}9){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{-2x+10}{5-4x-x^2}dx$

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

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

$2$

MAT$2691/101/0/2025$

$(1\mathrm{.}12){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x-1}{\sqrt[\phantom{2}]{x^2-4x+3}}dx$

Question $2$: $10$ Marks

Sketch the parabola $y=2x^2+5$ and the line $y=3x+4$ on the same set of axes. The segment cut off from the parabola $y=2x^2+5$ by the line $y=3x+4$ is rotated around the $x-$axis once. Determine the volume swept through.