INTEGRAL CALCULUS. CHAPTER I. INTEGRATION OF FUNCTIONS OF ONE VARIABLE. The fundamental formulæ to which all integrals are reduced are the following. (a) sdx an n + 1 1 log 2 a (a), Xta -1 1 dx dx (e) sin and (a’ – 2*)! (a* – ) dx |(x* + a')} + x = log (x* = a*) dx (8) x x– a) dx ) (? (a + 22 )} + a ar (i) Sdx a“ or sd & €4* log a - - اردو sec a 1 1 By simple algebraic transformations we may frequently put an integral into a shape in which one or other of the preceding formulæ is at once applicable. dx xn-1 dx nbx" log (a + bx;"). a + bxn nb da-r) 1 (1) Saxo = a + bor -S which is integrated by (c) or by (d) according as 4ac – 6? > 0 or <0. Hence we have dx X + 1 The integral Sta+ bx + cauty dw is reduced to dx ਦੇ 2 (-)" dx 1 or to c 6 4c according as the upper or lower sign of c is taken ; and these are of the forms (f) or (e) respectively. Hence dæ dr 23 = log {2x – 1 + 2 (v* – X – 1)"}. dw 5) may be split into (2x + p) dx si X* + px + 9 2 ° + px + 9 the first of which is integrable by (c) and the second by (6). Hence x da log (a + 25x + 2*)! 2 x + 1 = sin -1 a + (15) Sæt +260 + m2 X + b tan } tan-1 W + 1 2) 6 (a* – b)] lla? – bo)!) (2x – 1) da 3 (16) S · log (x2 + 2x + 3) 22 cos sin A tan-1 cos O + x sin e - cos 0 log (1 – 2w cos 0 + a'?). In this example the numerator may be readily split by observing that 1 = cos" 0 + sive. By multiplying the numerator and denominator of a fraction by the same quantity it may frequently be split into integrable parts or reduced to an integrable shape. dx x-2 (29) Sola+60+ copy) = S(aw-+62-" + c)? --S (ax-+62-2 + c)? which is of the same form as Sca (30) Sant 1-log{+++5+6)} da Therefore sin-' (2) dx (31) Se cont (32) Sta+ bx + cxo)? (33) Sta+bø + cx*) 2 (20x + b) 2 (2a + b) x dx €* X (34) The integral sdx (1 + x) can be split into 2 } 1 1 * (1 + x) and as the second term within the brackets is the differential d 1 of the first, it is equivalent to Sdx €". ; and therefore d x 1 + x de-1 (37) Silver - Site |