Indefinite Integral of e raised to a negative fraction

In summary, the formula for finding the indefinite integral of e raised to a negative fraction is ∫e^(a/x)dx = -xe^(a/x) + C, where a is a constant. To solve this type of integral, rewrite the expression as e^(a/x), use the given formula, and add a constant of integration. The constant of integration, represented by C, accounts for all possible solutions and allows for a general solution to be obtained. The negative sign in the formula indicates that the integral will result in a negative value due to the nature of the function e^(a/x). Special cases to consider include when the constant a is equal to 0 or 1.
  • #1
MathMan09
1
0

Homework Statement


Find the constant, c, that satisfies the following equation:


Homework Equations


The integral is from -infinity to infinity

1 = c [tex]\int[/tex] e ^ -|x|/2 *dx

The Attempt at a Solution



c = 1/4

I have the solution given to me, but I do not understand how to get the steps to the answer.
 
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  • #2
Find the integral of e^(-|x|/2) from 0 to infinity (where you can drop the absolute value). Compare that with the integral of the same function from -infinity to 0.
 

1. What is the formula for finding the indefinite integral of e raised to a negative fraction?

The formula for finding the indefinite integral of e raised to a negative fraction is ∫e^(a/x)dx = -xe^(a/x) + C, where a is a constant.

2. How do you solve an indefinite integral of e raised to a negative fraction?

To solve an indefinite integral of e raised to a negative fraction, you first need to rewrite the expression as e^(a/x), where a is a constant. Then, use the formula ∫e^(a/x)dx = -xe^(a/x) + C to integrate the expression and add a constant of integration, represented by C.

3. Can you explain the meaning of the constant of integration in the indefinite integral of e raised to a negative fraction?

The constant of integration, represented by C, is an arbitrary constant that is added at the end of the integration process. It accounts for all possible solutions to the original indefinite integral and allows for a general solution to be obtained.

4. What is the significance of the negative sign in the formula for the indefinite integral of e raised to a negative fraction?

The negative sign in the formula for the indefinite integral of e raised to a negative fraction indicates that the integral will result in a negative value. This is due to the nature of the function e^(a/x), which decreases as x increases.

5. Are there any special cases to consider when solving an indefinite integral of e raised to a negative fraction?

Yes, there are some special cases to consider when solving an indefinite integral of e raised to a negative fraction. For example, if the constant a is equal to 0, the integral will result in a value of -x + C. Additionally, if the constant a is equal to 1, the integral will result in a value of -e^(1/x) + C.

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