INTEGRAL CALCULUS. CHAPTER I. INTEGRATION OF FUNCTIONS OF ONE VARIABLE. THE fundamental formulæ to which all integrals are reduced are the following.. 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. which is integrated by (c) or by (d) according as 4ac – b2 >0 according as the upper or lower sign of c are of the forms (f) or (e) respectively. 4 c2 dx b 20 is taken; and these Hence log {2x + 1 + 2 (1 + x + x2)3}. = sin-1 w = log {2x-1+2 (x2 = x - ·1)}. = sin -1 dx (ax + b) The integral + pa + q the first of which is integrable by (c) and the second by (b). - cos log (1 - 2x cos 0 + x2)§. In this example the numerator may be readily split by observing that 1 cos" 0+ sin2 0. 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 which is of the same form as √ (a + bx + ca2)3 ́ dx x (1 + x + x2) dx x (x2 + 2x − 1) dx √(a + bx + cxo)* = 00 Therefore (a + bx + cx2)‡ ̄ (4ac − b2) (a + bx + cx2)3 (30) Sa (1 (31) Sæ (x2 (32) (33) (34) The integral fda can be split into (1 + x)2 2 (2a + bx) + x (1 + x)2) and as the second term within the brackets is the differential of the first, it is equivalent to fdx |