An improper integral is an integral where either the upper or lower limit of integration is infinite, or the integrand is not defined at one or more points within the interval of integration.
The integral is considered improper because the lower limit of integration is negative infinity, making it impossible to calculate the definite integral.
To solve an improper integral, you must first determine if it converges or diverges. If it converges, you can use the limit definition of an integral or any known integration techniques to evaluate it. If it diverges, you can use comparison tests or other convergence tests to determine the behavior of the integral.
The integral $\int_{-\infty}^{0}e^{-|x|}dx$ converges absolutely, meaning that the integral converges regardless of the sign of the integrand. This can be shown by using the limit definition of an integral and evaluating the integral as the upper limit approaches negative infinity.
No, the integral $\int_{-\infty}^{0}e^{-|x|}dx$ cannot be evaluated exactly. However, it can be approximated using numerical methods such as the trapezoidal rule or Simpson's rule.