4 3 \u00b2 8ln x 2 12 ln x 9 \u00b3 C Solution For d Since 3 x e x ln 3 We can use the

# 4 3 ² 8ln x 2 12 ln x 9 ³ c solution for d since 3

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4 3 ² 8(ln x ) 2 ° 12 ln x + 9 ³ + C: Solution: For ( d ) : Since 3 x = e x ln 3 We can use the formula Z 3 x cos xdx = Z e x ln 3 cos x dx = 1 4 e x ln 3 (sin x + ln x cos x ) + C: Solution: For ( d ) : Assume n ³ 2 and apply the Integration by Parts Rule: I n = Z dx sin n x = Z sin 2 x + cos 2 x sin n x dx = I n ° 2 + Z cos cos x sin n x dx = I n ° 2 ° 1 n ° 1 cos x sin n ° 1 x ° 1 n ° 1 I n ° 2 = n ° 2 n ° 1 I n ° 2 ° cos x ( n ° 1) sin n ° 1 x : Notice that by applying the substitution t = cos x we get I 1 = Z dx sin x = ° Z ° sin x dx 1 ° cos 2 x = ° Z dt 1 ° t 2 = 1 2 Z ° 1 t ° 1 ° 1 t + 1 ± dt = 1 2 ln ¸ ¸ ¸ ¸ 1 ° t 1 + t ¸ ¸ ¸ ¸ + C = 1 2 ln ¸ ¸ ¸ ¸ (1 ° cos x ) 2 sin 2 x ¸ ¸ ¸ ¸ + C = ln ¸ ¸ ¸ ¸ 1 ° cos x sin x ¸ ¸ ¸ ¸ + C = ln j csc x ° cot x j + C: Consequently, by the reduction formula I 3 = 1 2 I 1 ° cos x 2 sin 2 x = 1 2 ln j csc x ° cot x j ° cos x 2 sin 2 x + C: 3 Integration of Rational Functions Among the elementary functions, the rational functions played a special role in the development of integration techniques. One of the important features of rational functions is the fact that their inde°nite integrals can be expressed by elementary functions, and that means that theoretically, as long as we are able to e/ectuate related algebraic computations (like computations of roots of the denominator), we will also be able to °nd an exact formula for the integral. Moreover, many other types of integrals can 11
be reduced, by special substitutions, to integrals of rational functions. However, this is not the case for all elementary functions. There are many known functions with integrals that can not be expressed by elementary functions, i.e. it is impossible to express them by an analytic formula using basic elementary functions, arithmetic operations, their compositions and inverses. The following are just few well known integrals that can not be expressed by elementary functions: Z e ° x 2 dx; ; Z sin x 2 dx; Z cos 2 xdx Z sin x x dx; Z cos x x dx; Z dx ln x ; but there are many more. A rational function P ( x ) Q ( x ) , where P ( x ) and Q ( x ) are two polynomials, is called a proper rational function if deg P ( x ) < deg Q ( x ) : By using the division algorithm for polynomials, every rational function f ( x ) = P 1 ( x ) Q ( x ) can be represented as a sum of a polynomial and a proper rational function, i.e. f ( x ) = P 1 ( x ) Q ( x ) = M ( x ) + P ( x ) Q ( x ) ; where P 1 ( x ) = M ( x ) Q ( x ) + P ( x ) ; with P ( x ) being the remainder of the division of P 1 ( x ) by Q ( x ) , i.e. deg P ( x ) < deg Q ( x ) . We say that a proper rational function P ( x ) Q ( x ) is irreducible if P ( x ) and Q ( x ) have no common linear or quadratic factors. Lemma 3.1 Suppose that P ( x ) Q ( x ) is a proper rational function such that Q ( x ) = ( x ° ² ) k Q o ( x ) , where ² is not a root of Q o ( x ) and k 2 N . Then there exist a constant A and a polynomial P o ( x ) such that P ( x ) Q ( x ) = A ( x ° ² ) k + P o ( x ) ( x ° ² ) k ° 1 Q o ( x ) ; where deg P o ( x ) < deg Q ( x ) ° 1 .