65. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. To evaluate ∫ (4x⁶)/(x⁴ + 3x²) dx, the first step is to find the partial fraction decomposition of the integrand.

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First, examine the integrand \( \frac{4x^{6}}{x^{4} + 3x^{2}} \). Notice that both numerator and denominator are polynomials in terms of \( x \).

Check the degree of the numerator and denominator: the numerator has degree 6, and the denominator has degree 4. Since the degree of the numerator is higher, consider simplifying the integrand by polynomial division before attempting partial fraction decomposition.

Perform polynomial division of \( 4x^{6} \) by \( x^{4} + 3x^{2} \) to rewrite the integrand as a polynomial plus a proper rational function (where the numerator degree is less than the denominator degree).

After polynomial division, if the remaining rational function has a denominator that can be factored into linear or irreducible quadratic factors, then partial fraction decomposition can be applied to that part.

Therefore, the first step is not to directly find the partial fraction decomposition of the original integrand, but to simplify it first by polynomial division. This shows that the statement is false.

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주요 개념

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Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down rational functions into simpler fractions that are easier to integrate. It applies when the integrand is a ratio of polynomials where the degree of the numerator is less than the degree of the denominator. This method is not suitable if the numerator's degree is equal to or greater than the denominator's.

Polynomial Degree Comparison

Before applying partial fraction decomposition, compare the degrees of the numerator and denominator polynomials. If the numerator's degree is higher or equal, perform polynomial division first to rewrite the integrand as a polynomial plus a proper fraction. This step ensures the integrand fits the form required for partial fractions.

Integration of Rational Functions

Integrating rational functions often involves simplifying the integrand through algebraic manipulation such as polynomial division or partial fractions. Recognizing the appropriate method depends on the form of the integrand, which guides the integration strategy and helps avoid unnecessary or incorrect steps.

65. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.a. To evaluate ∫ (4x⁶)/(x⁴ + 3x²) dx, the first step is to find the partial fraction decomposition of the integrand.