# The Calculus

## Harold Davis

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1922 Excerpt: ...of tan x and of sec x. Any even power of tan x may be reduced to a sum of even powers of sec x (§ 108 (c)). This may be integrated by the method mentioned in §110. Another method of integrating any even power of sec x consists in making the substitution u = tan x. Since Au = sec2 x dx, and since even powers of sec x may be reduced to a sum of even powers of tan x (§ 108 (c) ), the integration becomes very simple, as is illustrated by Example 1 below. Vsin x cos3 x // /"COS3 X sin2 x cos3 x dx. 20. / sin3 x cos2 x dx. 21.;. „ dx. / J sin2 x 22. f'rdx. 23. /"sin3 x cos3 x dx. 24. /., dx. J sin2 x J J sui2 x cos4 x „, /"sin3 x dx /-sin3 x, „_ /"cos2 x, 25. / 26. /--s-dx. 27. /-v-r-dx. / cos x./ cos-1 x./ sin'" x 28. ysin4 x cos3 x dx. 29. ysin 4x cos 3xdx. 30. Jsin 3x cos 4xdx. 31. Verify formula 111 of Table IV, E. 32. Verify formula 112 of Table IV, E. 112. Elliptic and Other Integrals. If the only irrationality is vQ, where Q is a polynomial of the third or fourth degree, the integral is called an elliptic integral. While no treatment of these integrals is given here, they are treated briefly in tables of integrals, and their values have been computed in the form of tables. See Tables, V, D, E. Some idea of these quantities may be obtained by imagining some person ignorant of logarithms. Then f dx would be beyond his powers, and we should tell him "values of the integral J" (1/x) dk can be found tabulated," which is precisely what is done in tables of Napierian logarithms. Of course as little as possible is tabulated; other allied forms are reduced to those tabulated by means of special formulas, given in the tables. The discussion of such integrals, as well as of those in which Q is o...

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