The Calculus

Cover The Calculus
The Calculus
Ellery W Ellery Williams Davis
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-^o Vl - . 36 sin2 ^ > «^30o V5 - 2 sin2 a; Jo Vl^^ VT^^ 4. Show that n dx 1 p dx Jo 1 + 2 X cos <^ + x2 2 Jo 1 + 2 X COS (^ + x2 2 sin ' and explain what occurs when 0=0, and when ^ = ir/2. 5. Integrate the following general integrals; where /'(x) denotes the derivative of /(x). (a) j'/(x)/'(x)dx. Id) jr(2x)dx. (&) j"e*Ax)/'^x)dx. («) ij^^^' C ' ff\ ( f'{x)dx (c) J/'(cosx)sinxdx. U) J i + [/(x)]2* 6. "Verify the result of integrating sin^ x (?x by comijaring it with the integral of cos^ u du by means of the substitution u — 7r/2 — x. 7. Evaluate each of the following integrals : (a) I i &\n{u-\- v)dudv. (6) t I se'dtds. 8. Calculate the area ^, between the x-axis and the curve y = x^ — 9 x^ + 23 X — 15, from x = 1 to x = 3, by direct integration and also by Simp- son's Kule. Find the centroid (x, y) of the same area. VII, § 118] GENERAL EXERCISES 223 9. Proceed as in Ex. 8 for each of the following curves, between the limits stated below : (a) y = 1 + X — X- + x^ ; x = to a; = 2.

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