Problem 33 - Essential Calculus

Question

State the method you would use to evaluate each integral. Explain why you chose that method. Do not integrate. (a) \(\int \frac{x+1}{x^{2}+2 x-8} d x\) (b) \(\int \frac{7 x+4}{x^{2}+2 x-8} d x\) (c) \(\int \frac{4}{x^{2}+2 x+5} d x\)

Step-by-Step Solution

Answer

For integral (a), the method of partial fraction decomposition would be used. For integral (b), polynomial division followed by partial fractions would be advised. For integral (c), the appropriate approach would be to use a trigonometric substitution.

Step 1: Evaluate Integral (a)

The integrand \(\frac{x+1}{x^{2}+2 x-8}\) can be recognized as a rational function, which is a function presented as the ratio of two polynomials. The degree of the numerator is less than the degree of the denominator. This structure suggests that we use the method of partial fraction decomposition, because it's supposed to break down the original rational function into simpler fractions, which will be easier to integrate.

Step 2: Evaluate Integral (b)

The integrand \(\frac{7x+4}{x^{2}+2 x-8}\) is a rational function where again, the degree of the numerator is less than the degree of the denominator. However, the numerator isn’t simply ‘x’ plus a constant as in part (a). While we can still approach this using partial fractions, it is often easier in this case to first perform a polynomial division, followed by partial fractions if necessary.

Step 3: Evaluate Integral (c)

The integrand \(\frac{4}{x^{2}+2 x+5}\) is a rational function where the degree of the numerator '0' is less than the degree of the denominator '2'. Notice that this integrand is not simply 'x' plus a constant as in part (a), nor a linear polynomial in the numerator as in part (b). Additionally, the denominator can't be factored any further. We are dealing here with a standard form of an irreducible quadratic in the denominator, suggesting that a good approach here would be to use a trigonometric substitution.