Integrals of the form fdx x" (a + bx") can be rationalized, is an integer, by assuming a + bx" = x2, and, Ρ + is an integer, by assuming a + bx” = x" x1. when (1) (x − 1)3 x 1 (a+ba)ilog 7 3 5 (x − 1)2 + a}· bx)3 — a1 α -1 (x − 1)1 == COS (a2+a3)* _ 2 a2(a2+a3) + a2 } . [dæa2(a2+a3)3=2(a2+a°) 1{(a2+a®)° (6) 7 2 (a + bx) (a + bx 7 5 5 2 3 (a + bx) + 6a (a + bx) — a2 dx x2 (a + bx) ~ 363 5 (a + bx) 3 r If an integral be a function of several fractional powers of a, it may be rationalized by assuming a≈', being x = equal to the product of the denominators of the indices. When the integral involves also fractional powers of binomials, such as a + bx, it can be rationalized by assuming a + bx = x”, r being the product of the denominators of the indices. If the binomials be of the form a + ba", they may be reduced to the preceding form by assuming a" = y. If the function to be integrated involve (a + bæ ± cx2)} it may be reduced to the preceding forms, as Various functions can be rationalized by assumptions for which no general rule can be given: familiarity with the transformations to which different substitutions lead is the best way of acquiring a knowledge of the most convenient assumption in particular cases. |