Panphobia said:Homework Statement
[itex]\int_{-\infty}^{+\infty} e^{-\left|x\right|} \,dx[/itex]
The Attempt at a Solution
So I know you are supposed to split this integral up into two different ones, from (b to 0) and (0 to a) where b is approaching - infinity, and a is approaching +infinity, but how would I take that antiderivative? Since the absolute value has a piecewise derivative.
An improper integral is an integral where one or both of the limits of integration are infinite or where the integrand has a discontinuity within the interval.
Improper integrals arise in many real-world applications and are essential for solving certain mathematical problems. They also provide a way to evaluate integrals that would otherwise be impossible to solve using traditional methods.
To solve an improper integral with infinite limits, we first split the integral into two separate integrals: one from the lower limit to a finite number, and one from a finite number to the upper limit. Then, we take the limit of both integrals as the finite number approaches the infinite limit. If both limits exist and are finite, we add them together to get the final value of the improper integral.
If the integrand has a discontinuity within the interval, we can use the same process as in question 3. However, in this case, we may need to split the integral into multiple parts, with each part containing a different type of discontinuity. Then, we take the limit of each part as the finite number approaches the discontinuity point.
Yes, improper integrals can have a finite solution as long as the limit of the integral exists and is finite. In some cases, improper integrals may also diverge to infinity or may not have a defined solution.