Now here is an apparent inconsistency; for it would appear from (61) and (62) that sin-1 * +606-1 = Sande - Sen det = 0; (63) which result is not correct ; because sin- + cos->= : we must therefore have recourse to the accurate process of definite integration; and let us take both integrals between the same limits; say, between the limits 0 and x; then by (11), Art. 5, po dx fir-x70_N ren (64) sin Sin therefore by addition (66) which is a correct result; (61) and (62) therefore are not simultaneously true in the forms of indefinite integrals. 1. dx 32.) Integration of (2 ax — x2) "versih ā (2ax-x2)1' 36.] The relations existing between some of these last integrals deserve consideration; for taking the definite integral of (69) between the limits x and 0, we have * dx 1x +(a? + x2), (73) Jo (a? + x2) but the left-hand member may be put under the following form, whereby its integral is determined by equation (61): viz. and the identity of (73) and (74) may thus be proved : replacing sin (2 V-1) in terms of its exponential value as given a (75) a which shews that the two results though different in form are identical in value. Similarly might fx dx might 22 I 2 be put in the form 1 fr d (x v 1) -, and be integrated according to ✓-11. a? — (xv1)2' Article 15, and the result shewn to be equal to (aza + b2 +c) whereby the integral is reduced to one or other of the forms (76) or (77). . dx dix Ex. 1. c(1–2x+3co) • 20 39.] The integration of (a? — x2) dx, and of (a +x2) dx. In these cases it is convenient to employ the method of integration by parts given by the formula, Ju dv = wv – Svdu. (78) To determine /(a? – x2)+dx. |