Intergrating a derivative of absolute value in exponential

In summary, the conversation discusses an integral problem with a range from -∞ to ∞ and an integrand that is a derivative of an absolute value function in exponential form. The problem asks for any ideas on how to solve it, including potentially re-writing the integrand without the absolute value sign. Additionally, the conversation mentions the possibility of using a technique to integrate piecewise functions and inquires about the sign of the constant, α.
  • #1
myoikos
1
0
hey~ folks!

i'm in trouble with this integral problem.

the range of the integral is from -∞ to ∞ and it has x as independent value.

and the integrand is a derivative of absolute value in exponential;

∫[itex]^{∞}_{-∞}[/itex] (d/dx e[itex]^{-a|x|}[/itex])dx

any idea for the problem??
 
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  • #2
myoikos said:
hey~ folks!

i'm in trouble with this integral problem.

the range of the integral is from -∞ to ∞ and it has x as independent value.

and the integrand is a derivative of absolute value in exponential; e[itex]^{a|x|}[/itex]

any idea for the problem??
Welcome to Physics Forums.

What is the definition of the absolute value function? Can you think of a way to re-write the integrand without the absolute value sign?
 
  • #3
Use a technique like one might use when integrating other piecewise functions; that integral can be expressed as the sum of two definite integrals.
 
  • #4
Is the constant, α, positive ?
 

FAQ: Intergrating a derivative of absolute value in exponential

What is the derivative of absolute value in exponential?

The derivative of absolute value in exponential is a mathematical operation that involves finding the rate of change of a quantity represented by an exponential function with an absolute value. It is commonly used in calculus to calculate the slope of a curve at a specific point.

How do you integrate a derivative of absolute value in exponential?

To integrate a derivative of absolute value in exponential, you can use the properties of the exponential function and the chain rule. First, rewrite the absolute value as a piecewise function and then integrate each piece separately. Finally, combine the results to get the overall integral.

Can you give an example of integrating a derivative of absolute value in exponential?

Sure, let's say we have the function f(x) = |e^x|. To integrate this, we first rewrite it as f(x) = e^x for x ≥ 0 and f(x) = -e^x for x < 0. Then, we can use the rule for integrating e^x to get ∫f(x) dx = e^x for x ≥ 0 and ∫f(x) dx = -e^x for x < 0. Therefore, the overall integral is ∫f(x) dx = e^x + C for all values of x.

What are some real-life applications of integrating a derivative of absolute value in exponential?

One example is in physics, where this concept is used to calculate the displacement of a moving object with a changing velocity. It is also commonly used in economics to calculate the growth rate of a population or the value of an investment with compound interest.

Is it possible to integrate a derivative of absolute value in exponential without using calculus?

No, integrating a derivative of absolute value in exponential requires the use of calculus and its principles. It is a more advanced mathematical concept that cannot be solved using basic algebra or other methods.

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