13. The differential coefficient of any function can always be found by the use of the rules given in the former part of the book, but it is not so with the integral of any assigned function. We know, for example, that if m be any num ber, positive or negative, except – 1, then fa dx = Xm+1 m+1' but when m=- - 1 this is not true; in this case we have dx [d=loga. If however we had not previously defined the x. term logarithm, and investigated the properties of a logarithm, we should have been unable to state what function would 1 give as its differential coefficient. Thus we may find our x selves limited in our powers of integration from our not having given a name to every particular function and investigated its properties. In order to effect any proposed integration, it will often be necessary to use artifices which can only be suggested by practice. 14. We add a few miscellaneous examples. It should be noticed, that 1(x) and 4, (x) being any functions of x, dx [{$, (x) + 4, (x)} dx = [$, (x) die + fø, (x) dx, or at least the two expressions which we assert to be equal can only differ by a constant; for if we differentiate both we arrive at the same result, namely 4, (x) + (x). |