Solve ∫e^(1/x) / x^3 dx from -1 to 0

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In summary, the purpose of solving this integral is to find the area under the curve of the function e^(1/x) / x^3 from -1 to 0, which has various applications in statistics and physics. The integral can be solved using substitution, where u = 1/x and du = -1/x^2 dx. The lower limit of integration is -1 because the function is not defined at x=0. The function e^(1/x) / x^3 has significance in limits, probability, and as an example of a continuous but not differentiable function. While there are other methods, substitution is the most commonly used method for solving this integral.
  • #1
lasers
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∫e^(1/x) / x^3 dx ?

This has to be simple it's an improper integral from -1 to 0
How to solve?
 
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  • #2


Use integration by parts.
 
  • #3
You might want to try the substitution u =1/x first, then parts.
 

1. What is the purpose of solving this integral?

The purpose of solving this integral is to find the area under the curve of the function e^(1/x) / x^3 from -1 to 0. This can be used in various applications such as calculating probabilities in statistics or determining the work done in physics.

2. How do you solve this integral?

This integral can be solved using a technique known as substitution. Let u = 1/x, then du = -1/x^2 dx. Substituting this into the integral and making appropriate adjustments, the integral becomes ∫-e^u du from -1 to 0. This can then be solved using basic integration rules.

3. Why is the lower limit of integration -1 instead of 0?

The function e^(1/x) / x^3 is not defined at x=0, so the integral cannot be evaluated at this point. Therefore, the lower limit of integration is chosen to be a value close to 0, but not equal to it. In this case, -1 is a suitable lower limit that allows the integral to be evaluated.

4. What is the significance of the function e^(1/x) / x^3 in mathematics?

This function is commonly used in the study of limits and convergence in calculus. It is also used in probability and statistics to represent certain distributions. In addition, it is a well-known example of a function that is continuous but not differentiable at x=0.

5. Can this integral be solved using other methods?

Yes, there are alternative methods to solve this integral, such as using integration by parts or using a computer algebra system. However, substitution is the most straightforward and commonly used method for solving this type of integral.

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