Page:Calculus Made Easy.pdf/283

 (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) $$-\log_\epsilon (1 - x) + C$$.

(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 $$=10.85$$.