Page:Calculus Made Easy.pdf/283

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(17) ${\displaystyle \log(1+x)+C}$.

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

(1) Area ${\displaystyle =60}$; mean ordinate ${\displaystyle =10}$.

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

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

(4) Area ${\displaystyle =1.57}$; mean ordinate ${\displaystyle =0.5}$.

(5) ${\displaystyle 0.572}$, ${\displaystyle 0.0476}$.

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

(7) ${\displaystyle 1.25}$.

(8) ${\displaystyle 79.4}$.

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

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

(12) Arithmetical mean ${\displaystyle =9.5}$; quadratic mean ${\displaystyle =10.85}$.