Improper Integrals: Solve ∫-∞ to ∞ e^-|x| dx

• Panphobia
In summary, the conversation discusses the approach to solving the integral of e^-|x| and suggests splitting the integral into two separate integrals. One contributor also suggests writing the integral as e^x from -infinity to 0 and e^-x from 0 to infinity.
Panphobia

Homework Statement

$\int_{-\infty}^{+\infty} e^{-\left|x\right|} \,dx$

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.

Panphobia said:

Homework Statement

$\int_{-\infty}^{+\infty} e^{-\left|x\right|} \,dx$

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.

Hi Panphobia!

Do you see you can write it as ##\int_{-\infty}^0 e^x\,dx +\int_0^{\infty} e^{-x}\,dx##?

1 person
I totally missed that, but yea I see it now, thanks!

1. What is an improper integral?

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.

2. Why do we need to solve improper integrals?

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.

3. How do you solve an improper integral with infinite limits?

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.

4. What is the process for solving an improper integral with a discontinuous integrand?

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.

5. Can improper integrals have a finite solution?

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.

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