Tham khảo tài liệu 'stewart - calculus - early transcendentals 6e hq (thomson, 2008) episode 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 474 CHAPTER 7 TECHNIQUES OF INTEGRATION this equation f X 5 2 1 J X 2 X - 2dX - J u I - tttJ dx 2 In IX - 11 - In IX 2 I C To see how the method of partial fractions works in general let s consider a rational function .XX _ P x f x Q x where P and Q are polynomials. It s possible to express f as a sum of simpler fractions provided that the degree of P is less than the degree of Q. Such a rational function is called proper. Recall that if P x anXn a -1Xn 1 01X ao where an 0 then the degree of P is n and we write deg P n. If f is improper that is deg P deg Q then we must take the preliminary step of dividing Q into P by long division until a remainder R x is obtained such that deg R deg Q . The division statement is X _ P x R x f w Qt - sw QU where S and R are also polynomials. As the following example illustrates sometimes this preliminary step is all that is required. f X3 X VI EXAMPLE I Find I - dx. J X - 1 x x 2 x-1 p x .2 x x x x .2 x x 2x 2x 2 2 SOLUTION Since the degree of the numerator is greater than the degree of the denominator we first perform the long division. This enables us to write y XX dx y I X2 X 2 -I- - dx J X - 1 X - 1 X3 X 2 3 I 2 2x 2 ln IX 11 C The next step is to factor the denominator Q x as far as possible. It can be shown that any polynomial Q can be factored as a product of linear factors of the form ax b and irreducible quadratic factors of the form ax2 bx c where b2 - 4ac 0 . For instance if Q x X4 - 16 we could factor it as QX X2 - 4 X2 4 X - 2 X 2 X2 4 The third step is to express the proper rational function R x Q x from Equation 1 as a sum of partial fractions of the form A Ax B ax by or ax2 bx c j SECTION INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS 475 A theorem in algebra guarantees that it is always possible to do this. We explain the details for the four cases that occur. CASE I The denominator Q x is a product of distinct linear factors. This means that we can write Q x fl1 x b1 a2x b2 akx bk where no factor
