Question

Which of the following integrals are improper?

1) ∫(0 to [infinity]) x² dx
2) ∫(1 to 2) 1/x dx
3) ∫(0 to 1) eˣ dx
4) ∫(0 to π/2) sin(x) dx

Answer 1

In the given list, only the first integral, ∫(0 to [infinity]) x² dx, is an improper integral due to its infinite upper limit of integration.

Step-by-step explanation:

The student asked which of the following integrals are improper:

1. ∫(0 to [infinity]) x² dx
2. ∫(1 to 2) 1/x dx
3. ∫(0 to 1) eˣ dx
4. ∫(0 to π/2) sin(x) dx

An improper integral is one that has either an infinite limit of integration or an integrand that becomes infinite within the limits of integration. In this case:

1. The first integral ∫(0 to [infinity]) x² dx is improper because it has an infinite upper limit of integration.
2. The second integral ∫(1 to 2) 1/x dx is a proper integral since it does not have an infinite limit of integration and the function 1/x does not become infinite between 1 and 2.
3. The third integral ∫(0 to 1) eˣ dx is also a proper integral because it has finite limits of integration and the function eˣ is finite between 0 and 1.
4. The fourth integral ∫(0 to π/2) sin(x) dx is proper, as the integrand is finite within the finite integration limits.

Therefore, only the first integral is an improper integral.