3. Ja da x xn+1 Ju* log a dr = 2+1 {log x = 11}· 4. Se sine de = − 0 cos + sin 0. n+1) dx = √√ (mx + x2) +m log {√x + √√(m+x)}. 21. x dx = = x2 2 2 √/ (x − 1) sin mx cos nxdx= ax 2 X 7 - 1)3 +log {x + √√(x2 — 1)}. + 8 (x − 1)2 + x} . asin (m+n)x-(m+n) cos (m+n) x 2 ax ea2 a sin (m + 2 a2 + (m + n)2 (m—n) cos (m — n) x a2 + (m − n)2 Je* cosx dx = + fe* (cos 3x + 3 cos x) dx This may be obtained by putting sinx = 0. п cos mx cos nx dx π are zero if m and n are unequal integers, and = if 22 CHAPTER II. RATIONAL FRACTIONS. 16. WE proceed to the integration of such expressions as where A, B,... A', B',... are constants, so that both numerator and denominator are finite rational functions of x. If m be equal to n, or greater than n, we may by division reduce the preceding to the form of an integral function of x, and a fraction in which the numerator is of lower dimensions in x than the denominator. As the integral function of x can be integrated immediately, we may confine ourselves to the case of a fraction having its numerator at least one dimension lower than its denominator. In order to effect the integration we resolve the fraction into a series of more simple fractions called partial fractions, the possibility of which we proceed to demonstrate. U Let be a rational fraction which is to be resolved into a V series of partial fractions; suppose V a function of x of the nth degree, and U a function of x of the (n-1)th degree at most; we may without loss of generality suppose the coefficient of x in V to be unity. Suppose V = (x − a) (x − b)” (x2 − 2ax + a2 + B2) (x2 − 2yx + y2 +82), so that the equation V=0 has (1) one real root = a, (2) r equal real roots, each = b, (3) a pair of imaginary roots a ± ẞ √(− 1), (4) s pairs of imaginary roots, each being y±√(− 1). |