Page:Calculus Made Easy.pdf/283

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 $=10.85$.