Next, by substituting x = 2, x = 2, the equation reduces to 8 = B ( 2 ) ( 3 ), 8 = B ( 2 ) ( 3 ), or equivalently B = 4 / 3. For example, if we substitute x = 0, x = 0, the equation reduces to 2 = A ( −2 ) ( 1 ). Therefore, by choosing values of x x carefully and substituting them into the equation, we may find A, A, B, B, and C C easily. That is, this equation must be true for any value of x x we care to substitute into it. If the decomposition is set up correctly, then there must be values of A, A, B, B, and C C that satisfy Equation 3.8 for all values of x. The method of strategic substitution is based on the assumption that we have set up the decomposition correctly. The following example, although not requiring partial fraction decomposition, illustrates our approach to integrals of rational functions of the form ∫ P ( x ) Q ( x ) d x, ∫ P ( x ) Q ( x ) d x, where deg ( P ( x ) ) ≥ deg ( Q ( x ) ). We then do a partial fraction decomposition on R ( x ) Q ( x ). In the case when deg ( P ( x ) ) ≥ deg ( Q ( x ) ), deg ( P ( x ) ) ≥ deg ( Q ( x ) ), we must first perform long division to rewrite the quotient P ( x ) Q ( x ) P ( x ) Q ( x ) in the form A ( x ) + R ( x ) Q ( x ), A ( x ) + R ( x ) Q ( x ), where deg ( R ( x ) ) < deg ( Q ( x ) ). It is also extremely important to keep in mind that partial fraction decomposition can be applied to a rational function P ( x ) Q ( x ) P ( x ) Q ( x ) only if deg ( P ( x ) ) < deg ( Q ( x ) ). As we shall see, this form is both predictable and highly dependent on the factorization of the denominator of the rational function. The key to the method of partial fraction decomposition is being able to anticipate the form that the decomposition of a rational function will take. Using this method, we can rewrite an expression such as: 3 x x 2 − x − 2 3 x x 2 − x − 2 as an expression such as 1 x + 1 + 2 x − 2. In this section, we examine the method of partial fraction decomposition, which allows us to decompose rational functions into sums of simpler, more easily integrated rational functions.
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