Final answer:
If both integrals from a to infinity and from negative infinity to a of a function f(x) are convergent, then the integral from negative infinity to infinity of f(x) is defined as convergent. The correct answer is A.
Step-by-step explanation:
If both ∫[a to ∞] f(x)dx and ∫[-∞ to a] f(x)dx are convergent integrals, it means that the area under the curve f(x) from a to infinity and the area under the curve f(x) from negative infinity to a both exist and are finite. Therefore, when we define the integral from negative infinity to infinity of a function, we are essentially adding these two areas together.
According to the rules of calculus, if each of these integrals converges separately, their sum is also convergent. Thus, the correct answer here would be A) Convergent. This concept is similar to the idea presented in physics and engineering texts where a closed path line integral is the sum of integrals over two paths: one from A to B, and one from B to A, with the integral over one path being the negative of the other if traced in the opposite direction.
In conclusion, the integral of a function f(x) from negative infinity to infinity, denoted as ∫[-∞ to ∞] f(x)dx, is defined to be convergent if both the integral from a to infinity and the integral from negative infinity to a are convergent.