The Calculus and Its Applications; A Practical Treatise for Beginners, Especially Engineering Students (Paperback)

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1909 edition. Excerpt: ...constant. ExAMPLE 2.--Find sec2(7 + 4x)dx. Let 7 + 4x = z, then 4dx = dz, or (Z =-(%. the question becomes 2 secgzdz, which is itanz (see list of fundamental integrals), or item (7 + 4x), which is the answer required. ExAMPLE 3.--Find/e5+2"dx. Let 5 + 2.v = z, 2d: o = dz, and the given integral becomes; I' ezdz, which is % e' = %e5+2." Ans. No'.rE.--In a similar way it can be easily shown that f e''''"' = % e''+'1'', when p and q are constants.. 9.2:2--4x--6 ) ExAMPLE 4.--Find 3l----------x3 + 2302 _ 6x + 8 dx. Let 3x3+2x'--6x---8=z, then (9.1c2 + 4x--6)d.7J = dz. Hence the integral may be written /% = logz, = log(3a3--2r'--6.'c + 8).-Ans. N.B.--IMPORTANT RULE. If the numerator of the expression to be integrated is the digferential co-efiicient of the denominator, the integral required is the logarithm of the denominator. ExAMPLE 5.--Find/xdi b. Ans.: log(x--b). E 6 F' d j/Liixi A ' 2A/ b xAMPLE.--in Va/_ + b. ns...21; +.. dx 1 ExAMPLE 7.--Find _/(x + b)._, . Ans..--Z + b ExAMPLE 8.--Find /_ (6 + 5x) dx. Ans.: %(6 + 5.e)f. ExAMPLE 9.--Find sin(a + bx)da. Ans.: --51-cos(a + bx).. d ExAMPLE 10.--F1ndf(. Ans.: log(tan"1: ). sin(log x)dx ExAMPLE 11.--Find x Ans.: --cos(log x). ExAMPLE 12.--Find _c0i "id." (Let sin x = sin x Ans.: ---2 (sin x)--', _ 1 1 _' _ 71 sin477' ExAMPLE 13.--Find/(%xilwx)3. (Let 1 + x--_--z. Then dx: dz, x" = (z--1)2, and the integral becomes 1 2 _= log z + 2--, . Ans. III. Integration by the aid of Partial Fractions. We deal here with rational fractions only, i.e., those in which both numerator and denominator are rational integral functions of the variable or variables concerned. Such fractions...