November 2022 – Page 3

November 4, 2022December 14, 2022

202211041415 Solution to 1975-CE-AMATH-I-2

Find the values of $x$ satisfying both of the following inequalities:

$\begin{aligned} (1+x)(6-x)&\leqslant -8 \\ 3x-1 & \geqslant 5\\ \end{aligned}$

Roughwork.

The first inequality is

$\begin{aligned} (1+x)(6-x) & \leqslant -8 \\ 6+5x-x^2 \leqslant -8 \\ x^2-5x-14 \geqslant 0 \\ (x-7)(x+2) \geqslant 0 \\ x\leqslant -2\textrm{ \scriptsize{OR} }x\geqslant 7 \\ \end{aligned}$

whereas the second

$\begin{aligned} 3x-1 & \geqslant 5 \\ 3x & \geqslant 6 \\ x & \geqslant 2 \\ \end{aligned}$

In sum, $x\geqslant 7$ .

November 4, 2022November 4, 2022

202211041204 Solution to 2002-CE-AMATH-II-4

Find $\displaystyle{\int_{0}^{\frac{1}{2}}\frac{1}{\sqrt{1-x^2}}\,\mathrm{d}x}$ .
[Hint: Let $x=\sin\theta$ .]

Roughwork.

$\begin{aligned} &\quad\, \int_{0}^{\frac{1}{2}}\frac{1}{\sqrt{1-x^2}}\,\mathrm{d}x \\ \dots\enspace & \textrm{let }x=\sin\theta\textrm{ s.t. }\mathrm{d}x=\cos\theta\,\mathrm{d}\theta \enspace\dots \\ & = \int_{x=0}^{x=\frac{1}{2}}\frac{\cos\theta}{\sqrt{1-(\sin\theta )^2}}\,\mathrm{d}\theta \\ & = \int_{x=0}^{x=\frac{1}{2}}\frac{\cos\theta}{\cos\theta}\,\mathrm{d}\theta \\ & = \int_{x=0}^{x=\frac{1}{2}}\mathrm{d}\theta \\ & = \big[\theta \big]\big|_{x=0}^{x=\frac{1}{2}} \\ & = \big[\sin^{-1}x\big]\big|_{0}^{\frac{1}{2}} \\ & = \frac{\pi}{6}\\ \end{aligned}$

November 4, 2022November 4, 2022

202211041147 Solution to 2003-CE-AMATH-II-1

Find $\displaystyle{\int \cos^2\theta\,\mathrm{d}\theta}$ .

Roughwork.

We need the following

Lemma. (Double-angle formulae)

$\begin{aligned} \cos 2\theta & = \cos^2\theta -\sin^2\theta \\ & = \cos^2\theta - (1-\cos^2\theta )\\ & = 2\cos^2\theta -1 \\ \cos^2\theta & = \frac{1+\cos 2\theta}{2} \\ \end{aligned}$

Hence,

$\begin{aligned} &\quad\, \int\cos^2\theta\,\mathrm{d}\theta \\ & = \int \bigg( \frac{1+\cos 2\theta}{2}\bigg) \,\mathrm{d}\theta \\ & = \frac{1}{2}\int \mathrm{d}\theta + \frac{1}{2}\int \cos 2\theta\,\mathrm{d}\theta \\ & = \frac{\theta}{2} + \frac{1}{4}\sin 2\theta + C\textrm{ for some constant} \\ \end{aligned}$

November 4, 2022November 4, 2022

202211041133 Solution to 2004-CE-AMATH-II-1

Find

(a) $\displaystyle{\int\cos (3x+1)\,\mathrm{d}x}$ ,
(b) $\displaystyle{\int (2-x)^{2004}\,\mathrm{d}x}$ .

Roughwork.

(a)

$\begin{aligned} &\quad\, \int\cos (3x+1)\,\mathrm{d}x \\ \dots\enspace & \textrm{let }u=3x+1\textrm{ s.t. } \mathrm{d}x=\frac{\mathrm{d}u}{3}\enspace\dots \\ & = \int (\cos u)\bigg(\frac{\mathrm{d}u}{3}\bigg) \\ & = \frac{\sin u}{3}+C\textrm{ for some constant} \\ & = \frac{\sin (3x+1)}{3} + C \\ \end{aligned}$

(b)

$\begin{aligned} &\quad\, \int (2-x)^{2004}\,\mathrm{d}x \\ \dots\enspace &\textrm{let }u=2-x\textrm{ s.t. }\mathrm{d}x=-\mathrm{d}u \\ & = -\int u^{2004}\,\mathrm{d}u \\ & = -\frac{u^{2005}}{2005} + C\textrm{ for some constant} \\ & = -\frac{(2-x)^{2005}}{2005} + C \\ \end{aligned}$

November 4, 2022November 4, 2022

202211041123 Solution to 2004-CE-AMATH-II-3

The slope at any point $(x,y)$ of a curve $C$ is given by

$\displaystyle{\frac{\mathrm{d}y}{\mathrm{d}x}=3x^2+1}$ .

If the $x$ -intercept of $C$ is $1$ , find the equation of $C$ .

Roughwork.

$\begin{aligned} y & = \int\frac{\mathrm{d}y}{\mathrm{d}x}\,\mathrm{d}x \\ & = \int (3x^2+1)\,\mathrm{d}x \\ y(x) & = x^3 + x + C\textrm{ for some constant} \\ \end{aligned}$

Substituting $(1,0)$ for $(x,y(x))$ :

$\begin{aligned} 0 & = (1)^3+(1) + C \\ C & = -2 \\ \end{aligned}$

In conclusion, the equation of $C$ is

$y(x) = x^3 + x -2$ .

November 4, 2022November 4, 2022

202211041114 Solution to 2005-CE-AMATH-II-1

Find $\displaystyle{\int (2x-3)^7\,\mathrm{d}x}$ .

Roughwork.

$\begin{aligned} &\quad\, \int (2x-3)^7\,\mathrm{d}x \\ \dots\enspace & \textrm{let }u=2x-3\textrm{ s.t. }\mathrm{d}x=\frac{\mathrm{d}u}{2} \\ & = \int u^7\bigg( \frac{\mathrm{d}u}{2}\bigg) \\ & = \frac{1}{2} \int u^7\,\mathrm{d}u \\ & = \frac{1}{2}\bigg(\frac{u^8}{8}\bigg) +C\textrm{ for some constant} \\ & = \frac{u^8}{16} + C \\ & = \frac{(2x-3)^8}{16} +C \\ \end{aligned}$

November 4, 2022November 4, 2022

202211041009 Solution to 2005-CE-AMATH-II-10

(a) Show that

$\displaystyle{\frac{\mathrm{d}}{\mathrm{d}x}[x(x+1)^n]=(x+1)^{n-1}[(n+1)x+1]}$ ,

where $n$ is a rational number.
(b) The slope at any point $(x,y)$ of a curve $C$ is given by

$\displaystyle{\frac{\mathrm{d}y}{\mathrm{d}x}=(x+1)^{2004}(2006x+1)}$ .

If $C$ passes through the point $(-1,1)$ , find the equation of $C$ .

Roughwork.

(a)

Beginning by integration on the right hand side,

$\begin{aligned} &\quad\, \int (x+1)^{n-1}[(n+1)x+1]\,\mathrm{d}x \\ & = \int (x+1)^{n-1}[(n+1)(x+1)-n]\,\mathrm{d}x \\ & = (n+1)\int (x+1)^n\,\mathrm{d}x - n\int (x+1)^{n-1}\,\mathrm{d}x \\ & = (n+1)\frac{(x+1)^{n+1}}{n+1} - n\frac{(x+1)^{(n-1)+1}}{(n-1)+1} + C \\ & = (x+1)^{n+1}-(x+1)^n + C \\ & = (x+1)^n(x+1) - (x+1)^n + C \\ & = x(x+1)^n + C \textrm{ for some constant}\\ \end{aligned}$

or by differentiation on the left hand side,

$\begin{aligned} &\quad\,\frac{\mathrm{d}}{\mathrm{d}x}[x(x+1)^n] \\ & = x\cdot \frac{\mathrm{d}}{\mathrm{d}x}\big( (x+1)^n\big) + \frac{\mathrm{d}}{\mathrm{d}x} (x)\cdot (x+1)^n \\ & = x\cdot (n)(x+1)^{n-1}(1) + 1\cdot (x+1)^n \\ & = (nx)(x+1)^{n-1}+(x+1)(x+1)^{n-1} \\ & = (x+1)^{n-1}[(n+1)x+1] \\ \end{aligned}$

(b)

$\begin{aligned} y & = \int \frac{\mathrm{d}y}{\mathrm{d}x}\,\mathrm{d}x \\ & = \int (x+1)^{(2005)-1}[( (2005)+1)x+1] \\ y(x) & \stackrel{\textrm{(a)}}{=} x(x+1)^{2005}+C\textrm{ for some constant} \\ \end{aligned}$

Substituting $-1$ for $x$ and $1$ for $y(x)$ ,

$\begin{aligned} 1 & = (-1)((-1)+1)^{2005} + C \\ C & = 1 \\ \end{aligned}$

In conclusion, the equation of $C$ is

$y=x(x+1)^{2005}+1$ .

November 4, 2022November 4, 2022

202211041001 Solution to 2007-CE-AMATH-II-1

Find $\displaystyle{\int\frac{x^4+1}{x^2}\,\mathrm{d}x}$ .

Roughwork.

$\begin{aligned} &\quad\, \int\frac{x^4+1}{x^2}\,\mathrm{d}x\\ & = \int (x^2+x^{-2})\,\mathrm{d}x \\ & = \frac{x^3}{3} + \frac{x^{-1}}{(-1)} + C\textrm{ for some constant} \\ & = \frac{x^3}{3} - \frac{1}{x} + C \\ \end{aligned}$

November 4, 2022November 4, 2022

202211040951 Solution to 2008-CE-AMATH-II-1

Find $\displaystyle{\int (8x+5)^{250}\,\mathrm{d}x}$ .

Roughwork.

$\begin{aligned} &\quad\, \int (8x+5)^{250}\,\mathrm{d}x \\ \dots\enspace &\textrm{let }u=8x+5\textrm{ s.t. }\mathrm{d}x=\frac{\mathrm{d}u}{8}\enspace\dots \\ & = \int u^{250}\,\bigg(\frac{\mathrm{d}u}{8}\bigg) \\ & = \frac{1}{8}\bigg(\frac{u^{251}}{251}\bigg) +C\textrm{ for some constant} \\ & = \frac{(8x+5)^{251}}{2008}+C \\ \end{aligned}$

November 4, 2022November 4, 2022

202211040914 Solution to 2009-CE-AMATH-II-1

Find

(a) $\displaystyle{\int (4x+1)^2\,\mathrm{d}x}$ ,
(b) $\displaystyle{\int \sin 3\theta\cos\theta\,\mathrm{d}\theta}$ .

Roughwork.

(a)

$\begin{aligned} & \quad\, \int (4x+1)^2\,\mathrm{d}x \\ \dots\enspace & \textrm{let }u=4x+1\textrm{ s.t. }\mathrm{d}x=\frac{\mathrm{d}u}{4}\enspace\dots \\ & = \int u^2\,\bigg(\frac{\mathrm{d}u}{4}\bigg) \\ & = \frac{u^3}{12}+C \textrm{ for some constant}\\ & = \frac{(4x+1)^3}{12} + C \\ \end{aligned}$

(b)

$\begin{aligned} & \quad\, \int \sin 3\theta\cos\theta\,\mathrm{d}\theta \\ & = \int \bigg(\frac{e^{3\mathrm{i}\theta}-e^{-3\mathrm{i}\theta}}{2\mathrm{i}}\bigg)\bigg(\frac{e^{\mathrm{i}\theta}+e^{-\mathrm{i}\theta}}{2}\bigg)\,\mathrm{d}\theta \\ & = \int \frac{e^{4\mathrm{i}\theta}+e^{2\mathrm{i}\theta}-e^{-2\mathrm{i}\theta}-e^{-4\mathrm{i}\theta}}{4\mathrm{i}} \,\mathrm{d}\theta \\ & = \frac{1}{2}\int \bigg( \frac{e^{4\mathrm{i}\theta}-e^{-4\mathrm{i}\theta}}{2\mathrm{i}} + \frac{e^{2\mathrm{i}\theta}-e^{-2\mathrm{i}\theta}}{2\mathrm{i}} \bigg) \,\mathrm{d}\theta \\ & = \frac{1}{2}\int (\sin 4\theta +\sin 2\theta )\,\mathrm{d}\theta \\ & = \frac{1}{8}\int \sin 4\theta\,\mathrm{d}(4\theta ) + \frac{1}{4}\int\sin 2\theta\,\mathrm{d}(2\theta ) \\ & = -\frac{1}{8}\cos 4\theta - \frac{1}{4}\cos 2\theta +C\textrm{ for some constant} \\ \end{aligned}$