Calculus Made Easy/Answers to Exercises

by Silvanus Phillips Thompson

ANSWERS.

 

Exercises I. (p. 25.)

(1) ${\dfrac {dy}{dx}}=13x^{12}$.

(2) ${\dfrac {dy}{dx}}=-{\dfrac {3}{2}}x^{-{\frac {5}{2}}}$.

(3) ${\dfrac {dy}{dx}}=2ax^{(2a-1)}$.

(4) ${\dfrac {du}{dt}}=2.4t^{1.4}$.

(5) ${\dfrac {dz}{du}}={\dfrac {1}{3}}u^{-{\frac {2}{3}}}$.

(6) ${\dfrac {dy}{dx}}=-{\dfrac {5}{3}}x^{-{\frac {8}{3}}}$.

(7) ${\dfrac {du}{dx}}=-{\dfrac {8}{5}}x^{-{\frac {13}{5}}}$.

(8) ${\dfrac {dy}{dx}}=2ax^{a-1}$.

(9) ${\dfrac {dy}{dx}}={\dfrac {3}{q}}x^{\frac {3-q}{q}}$.

(10) ${\dfrac {dy}{dx}}=-{\dfrac {m}{n}}x^{-{\frac {m+n}{n}}}$.

 

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Exercises II. (p. 33.)

(1) ${\dfrac {dy}{dx}}=3ax^{2}$.

(2) ${\dfrac {dy}{dx}}=13\times {\frac {3}{2}}x^{\frac {1}{2}}$.

(3) ${\dfrac {dy}{dx}}=6x^{-{\frac {1}{2}}}$.

(4) ${\dfrac {dy}{dx}}={\dfrac {1}{2}}c^{\frac {1}{2}}x^{-{\frac {1}{2}}}$.

(5) ${\dfrac {du}{dz}}={\dfrac {an}{c}}z^{n-1}$.

(6) ${\dfrac {dy}{dt}}=2.36t$.

(7) ${\dfrac {dl_{t}}{dt}}=0.000012\times l_{0}$.

(8) ${\dfrac {dC}{dV}}=abV^{b-1}$, $0.98$, $3.00$ and $7.47$ candle power per volt respectively.

(9) ${\dfrac {dn}{dD}}=-{\dfrac {1}{LD^{2}}}{\sqrt {\dfrac {gT}{\pi \sigma }}},{\dfrac {dn}{dL}}=-{\dfrac {1}{DL^{2}}}{\sqrt {\dfrac {gT}{\pi \sigma }}}$.
$\;\quad {\dfrac {dn}{d\sigma }}=-{\dfrac {1}{2DL}}{\sqrt {\dfrac {gT}{\pi \sigma ^{3}}}},{\dfrac {dn}{dT}}={\dfrac {1}{2DL}}{\sqrt {\dfrac {g}{\pi \sigma T}}}$.

(10) ${\dfrac {\text{Rate of change of P when t varies}}{\text{Rate of change of P when D varies}}}=-{\dfrac {D}{t}}$.

(11) $2\pi$, $2\pi r$, $\pi l$, ${\tfrac {2}{3}}\pi rh$, $8\pi r$, $4\pi r^{2}$.

(12) ${\dfrac {dD}{dT}}={\dfrac {0.000012l_{t}}{\pi }}$.

 

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Exercises III. (p. 46.)

(1) (a) $1+x+{\dfrac {x^{2}}{2}}+{\dfrac {x^{3}}{6}}+{\dfrac {x^{4}}{24}}+\ldots$.

(b) $2ax+b$.

(c) $2x+2a$.

(d) $3x^{2}+6ax+3a^{2}$.

(2) ${\dfrac {dw}{dt}}=a-bt$.

(3) ${\dfrac {dy}{dx}}=2x$.

(4) $14110x^{4}-65404x^{3}-2244x^{2}+8192x+1379$.

(5) ${\dfrac {dx}{dy}}=2y+8$.

(6) $185.9022654x^{2}+154.36334$.

(7) ${\dfrac {-5}{(3x+2)^{2}}}$.

(8) ${\dfrac {6x^{4}+6x^{3}+9x^{2}}{(1+x+2x^{2})^{2}}}$.

(9) ${\dfrac {ad-bc}{(cx+d)^{2}}}$.

(10) ${\dfrac {anx^{-n-1}+bnx^{n-1}+2nx^{-1}}{(x^{-n}+b)^{2}}}$.

(11) $b+2ct$.

(12) $R_{0}(a+2bt)$, $R_{0}\left(a+{\dfrac {b}{2{\sqrt {t}}}}\right)$, $-{\dfrac {R_{0}(a+2bt)}{(1+at+bt^{2})^{2}}}$ or ${\dfrac {R^{2}(a+2bt)}{R_{0}}}$.

(13) $1.4340(0.000014t-0.001024)$, $-0.00117$, $-0.00107$, $-0.00097$.

(14) ${\dfrac {dE}{dl}}=b+{\dfrac {k}{i}}$, ${\dfrac {dE}{di}}=-{\dfrac {c+kl}{i^{2}}}$.

 

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Exercises IV. (p. 51.)

(1) $17+24x$; $24$.

(2) ${\dfrac {x^{2}+2ax-a}{(x+a)^{2}}}$; ${\dfrac {2a(a+1)}{(x+a)^{3}}}$

(3) $1+x+{\dfrac {x^{2}}{1\times 2}}+{\dfrac {x^{3}}{1\times 2\times 3}}$.

(4) (Exercises III.):

(1) (a) ${\dfrac {d^{2}y}{dx^{2}}}={\dfrac {d^{3}y}{dx^{3}}}=1+x+{\frac {1}{2}}x^{2}+{\frac {1}{6}}x^{3}+\ldots$

(b) $2a$, $0$.

(c) $2,0$.

(d) $6x+6a,6$.

(2) $-b$, $0$.

(3) $2$, $0$.

(4) $56440x^{3}-196212x^{2}-4488x+8192$.
$\quad \quad \quad 169320x^{2}-392424x-4488$.

(5) $2$, $0$.

(6) $371.80453x$, $371.80453$.

(7) ${\dfrac {30}{(3x+2)^{3}}}$, $-{\dfrac {270}{(3x+2)^{4}}}$.

(Examples, p. 41):

(1) ${\dfrac {6a}{b^{2}}}x$, ${\dfrac {6a}{b^{2}}}$.

(2) ${\dfrac {3a{\sqrt {b}}}{2{\sqrt {x}}}}-{\dfrac {6b{\sqrt[{3}]{a}}}{x^{3}}}$, ${\dfrac {18b{\sqrt[{3}]{a}}}{x^{4}}}-{\dfrac {3a{\sqrt {b}}}{4{\sqrt {x^{3}}}}}$.

(3) ${\dfrac {2}{\sqrt[{3}]{\theta ^{8}}}}-{\dfrac {1.056}{\sqrt[{5}]{\theta ^{11}}}}$, ${\dfrac {2.3232}{\sqrt[{5}]{\theta ^{16}}}}-{\dfrac {16}{3{\sqrt[{3}]{\theta ^{11}}}}}$.

(4) $810t^{4}-648t^{3}+479.52t^{2}-139.968t+26.64$.
$\quad \quad \quad 3240t^{3}-1944t^{2}+959.04t-139.968$.

(5) $12x+2$, $12$.

(6) $6x^{2}-9x$, $12x-9$.

(7) ${\begin{aligned}&{\dfrac {3}{4}}\left({\dfrac {1}{\sqrt {\theta }}}+{\dfrac {1}{\sqrt {\theta ^{5}}}}\right)+{\dfrac {1}{4}}\left({\dfrac {15}{\sqrt {\theta ^{7}}}}-{\dfrac {1}{\sqrt {\theta ^{3}}}}\right).\\&{\dfrac {3}{8}}\left({\dfrac {1}{\sqrt {\theta ^{5}}}}-{\dfrac {1}{\sqrt {\theta ^{3}}}}\right)-{\dfrac {15}{8}}\left({\dfrac {7}{\sqrt {\theta ^{9}}}}+{\dfrac {1}{\sqrt {\theta ^{7}}}}\right).\end{aligned}}$

 

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Exercises V. (p. 64.)

(2) $64$; $147.2$; and $0.32$ feet per second.

(3) ${\dot {x}}=a-gt$; ${\ddot {x}}=-g$.

(4) $45.1$ feet per second.

(5) $12.4$ feet per second per second. Yes.

(6) Angular velocity $=11.2$ radians per second; angular acceleration $=9.6$ radians per second per second.

(7) $v=20.4t^{2}-10.8$. $a=40.8t$. $172.8in./sec.$, $122.4in./sec^{2}$.

(8) $v={\dfrac {1}{30{\sqrt[{3}]{(t-125)^{2}}}}}$, $a=-{\dfrac {1}{45{\sqrt[{3}]{(t-125)^{5}}}}}$.

(9) $v=0.8-{\dfrac {8t}{(4+t^{2})^{2}}}$, $a={\dfrac {24t^{2}-32}{(4+t^{2})^{3}}}$, $0.7926$ and $0.00211$.

(10) $n=2$, $n=11$.

 

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Exercises VI. (p. 73.)

(1) ${\dfrac {x}{\sqrt {x^{2}+1}}}$.

(2) ${\dfrac {x}{\sqrt {x^{2}+a^{2}}}}$.

(3) $-{\dfrac {1}{2{\sqrt {(a+x)^{3}}}}}$.

(4) ${\dfrac {ax}{\sqrt {(a-x^{2})^{3}}}}$.

(5) ${\dfrac {2a^{2}-x^{2}}{x^{3}{\sqrt {x^{2}-a^{2}}}}}$.

(6) ${\dfrac {{\frac {3}{2}}x^{2}\left[{\frac {8}{9}}x\left(x^{3}+a\right)-\left(x^{4}+a\right)\right]}{(x^{4}+a)^{\frac {2}{3}}(x^{3}+a)^{\frac {3}{2}}}}$.

(7) ${\dfrac {2a\left(x-a\right)}{(x+a)^{3}}}$.

(8) ${\frac {5}{2}}y^{3}$.

(9) ${\dfrac {1}{(1-\theta ){\sqrt {1-\theta ^{2}}}}}$.

 

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Exercises VII. (p. 75.)

(1) ${\dfrac {dw}{dx}}={\dfrac {3x^{2}\left(3+3x^{3}\right)}{27\left({\frac {1}{2}}x^{3}+{\frac {1}{4}}x^{6}\right)^{3}}}$.

(2) ${\dfrac {dv}{dx}}=-{\dfrac {12x}{{\sqrt {1+{\sqrt {2}}+3x^{2}}}\left({\sqrt {3}}+4{\sqrt {1+{\sqrt {2}}+3x^{2}}}\right)^{2}}}$.

(3) ${\dfrac {du}{dx}}=-{\dfrac {x^{2}\left({\sqrt {3}}+x^{3}\right)}{\sqrt {\left[1+\left(1+{\dfrac {x^{3}}{\sqrt {3}}}\right)^{2}\right]^{3}}}}$.

 

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Exercises VIII. (p. 91.)

(2) $1.44$.

(4) ${\dfrac {dy}{dx}}=3x^{2}+3$; and the numerical values are: $3$, $3{\frac {3}{4}}$, $6$, and $15$.

(5) $\pm {\sqrt {2}}$.

(6) ${\dfrac {dy}{dx}}=-{\dfrac {4}{9}}{\dfrac {x}{y}}$. Slope is zero where $x=0$; and is $\mp {\dfrac {1}{3{\sqrt {2}}}}$ where $x=1$.

(7) $m=4$, $n=-3$.

(8) Intersections at $x=1$, $x=-3$. Angles $153^{\circ }\;26'$, $2^{\circ }\;28'$.

(9) Intersections at $x=3.57$, $x=3.50$. Angles $16^{\circ }\;16'$.

(10) $x={\tfrac {1}{3}}$, $y=2{\tfrac {1}{3}}$, $b=-{\tfrac {5}{3}}$.

 

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Exercises IX. (p. 109.)

(1) Min.: $x=0$, $y=0$; max.: $x=-2$, $y=-4$.

(2) $x=a$.

(4) $25{\sqrt {3}}$ square inches.

(5) ${\dfrac {dy}{dx}}=-{\dfrac {10}{x^{2}}}+{\dfrac {10}{(8-x)^{2}}}$; $x=4$; $y=5$.

(6) Max. for $x=-1$; min. for $x=1$.

(7) Join the middle points of the four sides.

(8) $r={\tfrac {2}{3}}R$, $r={\dfrac {R}{2}}$, no max.

(9) $r=R{\sqrt {\dfrac {2}{3}}}$, $r={\dfrac {R}{\sqrt {2}}}$, $r=0.8506R$.

(10) At the rate of ${\dfrac {8}{r}}$ square feet per second.

(11) $r={\dfrac {R{\sqrt {8}}}{3}}$.

(12) $n={\sqrt {\dfrac {NR}{r}}}$.

 

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Exercises X. (p. 118.)

(1) Max.: $x=-2.19$, $y=24.19$; min.: $x=1.52$, $y=-1.38$.

(2) ${\dfrac {dy}{dx}}={\dfrac {b}{a}}-2cx$; ${\dfrac {d^{2}y}{dx^{2}}}=-2c$; $x={\dfrac {b}{2ac}}$ (a maximum).

(3) (a) One maximum and two minima. (b) One maximum. ($x=0$; other points unreal.)

(4) Min.: $x=1.71$, $y=6.14$.

(5) Max: $x=-.5$, $y=4$.

(6) Max.: $x=1.414$, $y=1.7675$. Min.: $x=-1.414$, $y=1.7675$.

(7) Max.: $x=-3.565$, $y=2.12$. Min.: $x=+3.565$, $y=7.88$.

(8) $0.4N$, $0.6N$.

(9) $x={\sqrt {\dfrac {a}{c}}}$.

(10) Speed $8.66$ nautical miles per hour. Time taken $115.47$ hours. Minimum cost £$112$. $12s$.

(11) Max. and min. for $x=7.5$, $y=\pm 5.414$. (See example no. 10, here.)

(12) Min.: $x={\tfrac {1}{2}}$, $y=0.25$; max.: $x=-{\tfrac {1}{3}}$, $y=1.408$.

 

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Exercises XI. (p. 130.)

(1) ${\dfrac {2}{x-3}}+{\dfrac {1}{x+4}}$.

(2) ${\dfrac {1}{x-1}}+{\dfrac {2}{x-2}}$.

(3) ${\dfrac {2}{x-3}}+{\dfrac {1}{x+4}}$.

(4) ${\dfrac {5}{x-4}}-{\dfrac {4}{x-3}}$.

(5) ${\dfrac {19}{13(2x+3)}}-{\dfrac {22}{13(3x-2)}}$.

(6) ${\dfrac {2}{x-2}}+{\dfrac {4}{x-3}}-{\dfrac {5}{x-4}}$.

(7) ${\dfrac {1}{6(x-1)}}+{\dfrac {11}{15(x+2)}}+{\dfrac {1}{10(x-3)}}$.

(8) ${\dfrac {7}{9(3x+1)}}+{\dfrac {71}{63(3x-2)}}-{\dfrac {5}{7(2x+1)}}$.

(9) ${\dfrac {1}{3(x-1)}}+{\dfrac {2x+1}{3(x^{2}+x+1)}}$.

(10) $x+{\dfrac {2}{3(x+1)}}+{\dfrac {1-2x}{3(x^{2}-x+1)}}$.

(11) ${\dfrac {3}{(x+1)}}+{\dfrac {2x+1}{x^{2}+x+1}}$.

(12) ${\dfrac {1}{x-1}}-{\dfrac {1}{x-2}}+{\dfrac {2}{(x-2)^{2}}}$.

(13) ${\dfrac {1}{4(x-1)}}-{\dfrac {1}{4(x+1)}}+{\dfrac {1}{2(x+1)^{2}}}$.

(14) ${\dfrac {4}{9(x-1)}}-{\dfrac {4}{9(x+2)}}-{\dfrac {1}{3(x+2)^{2}}}$.

(15) ${\dfrac {1}{x+2}}-{\dfrac {x-1}{x^{2}+x+1}}-{\dfrac {1}{(x^{2}+x+1)^{2}}}$.

(16) ${\dfrac {5}{x+4}}-{\dfrac {32}{(x+4)^{2}}}+{\dfrac {36}{(x+4)^{3}}}$.

(17) ${\dfrac {7}{9(3x-2)^{2}}}+{\dfrac {55}{9(3x-2)^{3}}}+{\dfrac {73}{9(3x-2)^{4}}}$.

(18) ${\dfrac {1}{6(x-2)}}+{\dfrac {1}{3(x-2)^{2}}}-{\dfrac {x}{6(x^{2}+2x+4)}}$.

 

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Exercises XII. (p. 153.)

(1) $ab(\epsilon ^{ax}+\epsilon ^{-ax})$.

(2) $2at+{\frac {2}{t}}$.

(3) $\log _{\epsilon }n$.

(5) $npv^{n-1}$.

(6) ${\frac {n}{x}}$.

(7) ${\frac {3\epsilon ^{-{\frac {x}{x-1}}}}{(x-1)^{2}}}$.

(8) $6x\epsilon ^{-5x}-5(3x^{2}+1)\epsilon ^{-5x}$.

(9) ${\frac {ax^{a-1}}{x^{a}+a}}$.

(10) $\left({\frac {6x}{3x^{2}-1}}+{\frac {1}{2\left({\sqrt {x}}+x\right)}}\right)\left(3x^{2}-1\right)\left({\sqrt {x}}+1\right)$.

(11) ${\frac {1-\log _{\epsilon }\left(x+3\right)}{\left(x+3\right)^{2}}}$.

(12) $a^{x}\left(ax^{a-1}+x^{a}\log _{\epsilon }a\right)$.

(14) Min.: $y=0.7$ for $x=0.694$.

(15) ${\frac {1+x}{x}}$.

(16) ${\frac {3}{x}}(\log _{\epsilon }ax)^{2}$.

 

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Exercises XIII. (p. 162.)

(1) Let ${\frac {t}{T}}=x$ (∴ $t=8x$), and use the Table on page 159.

(2) $T=34.627$; $159.46$ minutes.

(3) Take $2t=x$; and use the Table on page 159.

(5) (a) $x^{x}\left(1+\log _{\epsilon }x\right)$; (b) $2x(\epsilon ^{x})^{x}$; (c) $\epsilon ^{x^{x}}\times x^{x}\left(1+\log _{\epsilon }x\right)$.

(6) $0.14$ second.

(7) (a) $1.642$; (b) $15.58$.

(8) $\mu =0.00037$, $31^{m}{\frac {1}{4}}$.

(9) $i$ is $63.4\%$ of $i_{0}$, $220$ kilometres.

(10) $0.133$, $0.145$ $0.155$, mean $0.144$; $-{10.2}\%$, $-{0.9}\%$, $+{77.2}\%$.

(11) Min. for $x={\frac {1}{\epsilon }}$.

(12) Max. for $x=\epsilon$.

(13) Min. for $x=\log _{\epsilon }a$.

 

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Exercises XIV. (p. 173.)

(1) (i) ${\dfrac {dy}{d\theta }}=A\cos \left(\theta -{\dfrac {\pi }{2}}\right)$;

(ii) ${\dfrac {dy}{d\theta }}=2\sin \theta \cos \theta =\sin 2\theta$ and ${\dfrac {dy}{d\theta }}=2\cos 2\theta$;

(iii) ${\dfrac {dy}{d\theta }}=3\sin ^{2}\theta \cos \theta$ and ${\dfrac {dy}{d\theta }}=3\cos 3\theta$.

(2) $\theta =45^{\circ }$ or ${\dfrac {\pi }{4}}$ radians.

(3) ${\dfrac {dy}{dt}}=-n\sin 2\pi nt$.

(4) $a^{x}\log _{\epsilon }a\cos a^{x}$.

(5) ${\dfrac {\cos x}{\sin x}}={\text{cotan}}\;x$.

(6) $18.2\cos \left(x+26^{\circ }\right)$.

(7) The slop is ${\dfrac {dy}{d\theta }}=100\cos \left(\theta -15^{\circ }\right)$, which is a maximum when $(\theta -15^{\circ })=0$ or $\theta =15^{\circ }$; the value of the slope being then $=100$. When $\theta =75^{\circ }$ the slope is $100\cos(75^{\circ }-15^{\circ })=100\cos 60^{\circ }=100\times {\frac {1}{2}}=50$.

(8) $\cos \theta \sin 2\theta +2\cos 2\theta \sin \theta =2\sin \theta \left(\cos ^{2}\theta +\cos 2\theta \right)$

$=2\sin \theta \left(3\cos ^{2}\theta -1\right)$.

(9) $amn\theta ^{n-1}\tan ^{m-1}\left(\theta ^{n}\right)\sec ^{2}\theta ^{n}$.

(10) $\epsilon ^{x}\left(\sin ^{2}x+\sin 2x\right)$; $\epsilon ^{x}\left(\sin ^{2}x+2\sin 2x+2\cos 2x\right)$.

(11) (i) ${\dfrac {dy}{dx}}={\dfrac {ab}{\left(x+b\right)^{2}}}$;

(ii) ${\dfrac {a}{b}}\epsilon ^{-{\frac {x}{b}}}$;

(iii) ${\dfrac {1}{90^{\circ }}}\times {\dfrac {ab}{\left(b^{2}+x^{2}\right)}}$.

(12) (i) ${\dfrac {dy}{dx}}=\sec x\tan x$;

(ii) ${\dfrac {dy}{dx}}=-{\dfrac {1}{\sqrt {1-x^{2}}}}$;

(iii) ${\dfrac {dy}{dx}}={\dfrac {1}{1+x^{2}}}$;

(iv) ${\dfrac {dy}{dx}}={\dfrac {1}{x{\sqrt {x^{2}-1}}}}$;

(v) ${\dfrac {dy}{dx}}={\dfrac {{\sqrt {3\sec x}}\left(3\sec ^{2}x-1\right)}{2}}$.

(13) ${\dfrac {dy}{d\theta }}=4.6\left(2\theta +3\right)^{1.3}\cos \left(2\theta +3\right)^{2.3}$.

(14) ${\dfrac {dy}{d\theta }}=3\theta ^{2}+3\cos \left(\theta +3\right)-\log _{\epsilon }3\left(\cos \theta \times 3^{\sin \theta }+3\theta \right)$.

(15) $\theta =\cot \theta ;\theta =\pm 0.86$; is max. for $+\theta$, min. for $-\theta$.

 

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Exercises XV. (p. 180.)

(1) $x^{3}-6x^{2}y-2y^{2}$; ${\frac {1}{3}}-2x^{3}-4xy$.

(2) $2xyz+y^{2}z+z^{2}y+2xy^{2}z^{2}$;

$2xyz+x^{2}z+xz^{2}+2x^{2}yz^{2}$;

$2xyz+x^{2}y+xy^{2}+2x^{2}y^{2}z$.

(3) ${\dfrac {1}{r}}\{\left(x-a\right)+\left(y-b\right)+\left(z-c\right)\}={\dfrac {\left(x+y+z\right)-\left(a+b+c\right)}{r}}$; ${\dfrac {3}{r}}$.

(4) $dy=vu^{v-1}\,du+u^{v}\log _{\epsilon }u\,dv$.

(5) $dy=3\sin vu^{2}\,du+u^{3}\cos v\,dv$,

$dy=u\sin x^{u-1}\cos x\,dx+(\sin x)^{u}\log _{\epsilon }\sin xdu$,

$dy={\dfrac {1}{v}}\,{\dfrac {1}{u}}\,du-\log _{\epsilon }u{\dfrac {1}{v^{2}}}\,dv$.

(7) Minimum for $x=y=-{\tfrac {1}{2}}$.

(8) (a) Length $2$ feet, width = depth = $1$ foot, vol. = $2$ cubic feet.

(b) Radius = $2\pi$ feet = $7.46$ in., length = $2$ feet, vol. = $2.54$.

(9) All three parts equal; the product is maximum.

(10) Minimum for $x=y=1$.

(11) Min.: $x={\tfrac {1}{2}}$ and $y=2$.

(12) Angle at apex $=90^{\circ }$; equal sides = length =${\sqrt[{3}]{2V}}$.

 

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Exercises XVI. (p. 190.)

(1) $1{\tfrac {1}{3}}$.

(2) $0.6344$.

(3) $0.2624$.

(4) (a) $y={\tfrac {1}{8}}x^{2}+C$;

(b) $y=\sin x+C$.

(5) $y=x^{2}+3x+C$.

 

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Exercises XVII. (p. 205.)

(1) ${\dfrac {4{\sqrt {a}}x^{\frac {3}{2}}}{3}}+C$.

(2) $-{\dfrac {1}{x^{3}}}+C$.

(3) ${\dfrac {x^{4}}{4a}}+C$.

(4) ${\tfrac {1}{3}}x^{3}+ax+C$.

(5) $-2x^{-{\frac {5}{2}}}+C$.

(6) $x^{4}+x^{3}+x^{2}+x+C$.

(7) ${\dfrac {ax^{2}}{4}}+{\dfrac {bx^{3}}{9}}+{\dfrac {cx^{4}}{16}}+C$.

(8) ${\dfrac {x^{2}+a}{x+a}}=x-a+{\dfrac {a^{2}+a}{x+a}}$ by division. Therefore the answer is ${\dfrac {x^{2}}{2}}-ax+(a^{2}+a)\log _{\epsilon }(x+a)+C$. (See pages 199 and 201.)

(9) ${\dfrac {x^{4}}{4}}+3x^{3}+{\dfrac {27}{2}}x^{2}+27x+C$.

(10) ${\dfrac {x^{3}}{3}}+{\dfrac {2-a}{2}}x^{2}-2ax+C$.

(11) $a^{2}(2x^{\frac {3}{2}}+{\tfrac {9}{4}}x^{\frac {4}{3}})+C$.

(12) $-{\tfrac {1}{3}}\cos \theta -{\tfrac {1}{6}}\theta +C$.

(13) ${\dfrac {\theta }{2}}+{\dfrac {\sin 2a\theta }{4a}}+C$.

(14) ${\dfrac {\theta }{2}}-{\dfrac {\sin 2\theta }{4}}+C$.

(15) ${\dfrac {\theta }{2}}-{\dfrac {\sin 2a\theta }{4a}}+C$.

(16) ${\tfrac {1}{3}}\epsilon ^{3x}$.

(17) $\log(1+x)+C$.

(18) $-\log _{\epsilon }(1-x)+C$.

 

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Exercises XVIII. (p. 224.)

(1) Area $=60$; mean ordinate $=10$.

(2) Area $={\frac {2}{3}}$ of $a\times 2a{\sqrt {a}}$.

(3) Area $=2$; mean ordinate $={\dfrac {2}{\pi }}=0.637$.

(4) Area $=1.57$; mean ordinate $=0.5$.

(5) $0.572$, $0.0476$.

(6) Volume $=\pi r^{2}{\dfrac {h}{3}}$.

(7) $1.25$.

(8) $79.4$.

(9) Volume $=4.9348$; area of surface $=12.57$ (from $0$ to $\pi$).

(10) $a\log _{\epsilon }a$, ${\dfrac {a}{a-1}}\log _{\epsilon }a$.

(12) Arithmetical mean $=9.5$; quadratic mean $=10.85$.

(13) Quadratic mean $={\tfrac {1}{\sqrt {2}}}{\sqrt {A_{1}^{2}+A_{3}^{2}}}$; arithmetical mean $=0$. The first involves a somewhat difficult integral, and may be stated thus: By definition the quadratic mean will be

${\sqrt {{\dfrac {1}{2\pi }}\int _{0}^{2\pi }(A_{1}\sin x+A_{3}\sin 3x)^{2}\,dx}}$.

Now the integration indicated by

$\int (A_{1}^{2}\sin ^{2}x+2A_{1}A_{3}\sin x\sin 3x+A_{3}^{2}\sin ^{2}3x)\,dx$

is more readily obtained if for $\sin ^{2}x$ we write

${\dfrac {1-\cos 2x}{2}}$.

For $2\sin x\sin 3x$ we write $\cos 2x-\cos 4x$; and, for $\sin ^{2}3x$,

${\dfrac {1-\cos 6x}{2}}$.

Making these substitutions, and integrating, we get (see p. 202)

${\dfrac {A_{1}^{2}}{2}}\left(x-{\dfrac {\sin 2x}{2}}\right)+A_{1}A_{3}\left({\dfrac {\sin 2x}{2}}-{\dfrac {\sin 4x}{4}}\right)+{\dfrac {A_{3}^{2}}{2}}\left(x-{\dfrac {\sin 6x}{6}}\right)$.

At the lower limit the substitution of $0$ for $x$ causes all this to vanish, whilst at the upper limit the substitution of $2\pi$ for $x$ gives $A_{1}^{2}\pi +A_{3}^{2}\pi$. And hence the answer follows.

(14) Area is $62.6$ square units. Mean ordinate is $10.42$.

(16) $436.3$. (This solid is pear shaped.)

 

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Exercises XIX. (p. 233.)

(1) ${\dfrac {x{\sqrt {a^{2}-x^{2}}}}{2}}+{\dfrac {a^{2}}{2}}\sin ^{-1}{\dfrac {x}{a}}+C$.

(2) ${\dfrac {x^{2}}{2}}(\log _{\epsilon }x-{\tfrac {1}{2}})+C$.

(3) ${\dfrac {x^{a+1}}{a+1}}\left(\log _{\epsilon }x-{\dfrac {1}{a+1}}\right)+C$.

(4) $\sin \epsilon ^{x}+C$.

(5) $\sin(\log _{\epsilon }x)+C$.

(6) $\epsilon ^{x}(x^{2}-2x+2)+C$.

(7) ${\dfrac {1}{a+1}}(\log _{\epsilon }x)^{a+1}+C$.

(8) $\log _{\epsilon }(\log _{\epsilon }x)+C$.

(9) $2\log _{\epsilon }(x-1)+3\log _{\epsilon }(x+2)+C$.

(10) ${\frac {1}{2}}\log _{\epsilon }(x-1)+{\frac {1}{5}}\log _{\epsilon }(x-2)+{\frac {3}{10}}\log _{\epsilon }(x+3)+C$.

(11) ${\dfrac {b}{2a}}\log _{\epsilon }{\dfrac {x-a}{x+a}}+C$.

(12) $\log _{\epsilon }{\dfrac {x^{2}-1}{x^{2}+1}}+C$.

(13) ${\frac {1}{4}}\log _{\epsilon }{\dfrac {1+x}{1-x}}+{\frac {1}{2}}\arctan x+C$.

(14) ${\dfrac {1}{\sqrt {a}}}\log _{\epsilon }{\dfrac {{\sqrt {a}}-{\sqrt {a-bx^{2}}}}{x{\sqrt {a}}}}$. (Let ${\dfrac {1}{x}}=v$; then, in the result, let ${\sqrt {v^{2}-{\dfrac {b}{a}}}}=v-u$.)

You had better differentiate now the answer and work back to the given expression as a check.

Every earnest student is exhorted to manufacture more examples for himself at every stage, so as to test his powers. When integrating he can always test his answer by differentiating it, to see whether he gets back the expression from which he started.

There are lots of books which give examples for practice. It will suffice here to name two: R. G. Blaine’s The Calculus and its Applications, and F. M. Saxelby’s A Course in Practical Mathematics.

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