Question

Using the method of partial fractions, we wish to compute ∫ 03ˣ²−13x+401​ dx

(i) We begin by factoring the denominator of the rational function to obtain:
ˣ²−13x+40=(x−a)(x−b) for a

Answer 1

The student is asked to compute a definite integral using the method of partial fractions, which involves factoring a quadratic function and integrating the resulting partial fractions.

Step-by-step explanation:

The question pertains to the computation of a definite integral using the method of partial fractions. The integral in question is ∫ ⅛ dx. The given integrand needs to be expressed as a rational function whose denominator is factorable. Hence, we start by finding the factors of the denominator x² - 13x + 40, such that it can be expressed as (x - a)(x - b). To find the roots a and b, we can apply the quadratic formula to the quadratic equation x² - 13x + 40 = 0 which yields values based on the coefficients. Once the factors are identified, the rational function is decomposed into partial fractions, enabling us to integrate each term separately.

After integrating, we evaluate the antiderivative at the upper and lower limits of integration to calculate the definite integral's value. Remember that these solutions successfully factoring the quadratic expression and performing the integration correctly.