Question

Choose the most appropriate option. \[ \int_{-2}^{2} \frac{3x^7 - 2x^5 + x^3 - 3}{x^4 + 3x^2 + 1} \, dx \]

• A. does not exist
• B. 3
• C. \(\frac{1}{e}\)
• D. \(\infty\)

Hint:

Always check for discontinuities or singularities within the limits of integration for improper integrals.

Answer 1

The Correct Option is A

This is an improper integral because the function inside the integral is not continuous at \(x = 0\). When a function has a discontinuity or a singularity at a point within the domain of integration, it makes the integral improper. In such cases, the integral may not converge to a finite value in the standard sense. To evaluate this type of improper integral, one typically needs to consider limits or a more advanced method, such as principal value, depending on the nature of the singularity.

Since the function is not continuous at \(x = 0\), the integral does not exist in the standard sense without further clarification or regularization techniques to handle the discontinuity.