Final answer:
The given integrals are improper due to various reasons such as infinite limits of integration and singularities. We can evaluate these integrals as limits by substituting finite values for the limits and taking appropriate limits.
Step-by-step explanation:
a) The integral ∫e^(-x) dx from 0 to ∞ is improper because the upper limit of integration is infinity, which is not a finite number. This means that the area under the curve extends infinitely and cannot be calculated exactly. However, we can evaluate this integral as a limit by substituting a finite value for the upper limit, such as M, and taking the limit as M approaches infinity.
b) The integral ∫1/(x^2 + 1) dx from -∞ to ∞ is improper because it has two infinite limits of integration. Similar to the previous integral, we can evaluate this integral as a limit by substituting finite values for the limits, such as -M and M, and taking the limit as M approaches infinity.
c) The integral ∫1/x dx from 1 to ∞ is improper because the lower limit of integration is less than the upper limit. This violates the intuitive notion of integration as finding the area under a curve. We can evaluate this integral as a limit by substituting a finite value for the lower limit, such as a, and taking the limit as a approaches 0 from the right.
d) The integral ∫1/x dx from -1 to 1 is improper because there is a singularity at x = 0. The function 1/x is not defined at x = 0, so we have to split the interval into two parts: -1 to 0 and 0 to 1. Each part will have its own improper integral, and we can evaluate them separately.