Question

Calculating the integral of a function means calculating the area between its curve and the x-axis, in order to assign positive values where the function is positive and negative otherwise. However, we cannot take every function as integrable in an interval [a, b], because, before calculating the defined integral, we need to analyze the continuity of the function.

Considering this information, analyze the following assertions and the proposed relationship between them.

I. It is possible to calculate the integral of the function f (x) = (x²-9) / (x + 3), whose domain set is D = [-6.0].

Because:

II. The function can be simplified if the notable product f (x) = (x-3) (x + 3) / (x + 3) is performed, so that f (x) = x-3, being then a function defined in the whole range [-6,0] and, integrating, we have the primitive F (x) = x² / 2 - 3x + C and, calculating the definite integral, we have F (0) - F (-6) = 0 - 0 + C - (18 + 18 + C) = -36.

Next, check the correct alternative.

a) Assertion I is a true proposition, and II is a false proposition.

b) Assertions I and II are false propositions.

c) Assertion I is a false proposition, and II is a true proposition.

d) Assertions I and II are true propositions, but II is not a correct justification for I.

e) Assertions I and II are true propositions, and II is a correct justification for I.

Answer 1

Option (e) is true

Because:

The function can be simplified if the notable product f (x) = (x-3) (x + 3) / (x + 3) is performed, so that f (x) = x-3, being then a function defined in the whole range [-6,0] and, integrating, we have the primitive F (x) = x² / 2 - 3x + C and, calculating the definite integral, we have F (0) - F (-6) = 0 - 0 + C - (18 + 18 + C) = -36.

Hence we can calculate the integral of the function f (x) = (x²-9) / (x + 3), whose domain set is D = [-6.0].