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Magazine Chapter 7: Problem 46
Evaluate the integrals. \begin{equation}\int \frac{e^{-1 / x^{2}}}{x^{3}} d x\end{equation}
Short Answer
Expert verified
The integral evaluates to \( \frac{1}{2} e^{-1 / x^2} + C \).
Step by step solution
01
Identify the Integral Type
The given integral is \(\int \frac{e^{-1 / x^2}}{x^3} \, dx\). To solve this, we should recognize that it may be suitable for a substitution method because of the complexity in the exponent.
02
Choose Substitution Variable
Let's choose a substitution variable to simplify the integral. We can set \( u = -\frac{1}{x^2} \). This substitution is beneficial because it simplifies the exponent.
03
Compute the Derivative of the Substitution
Differentiate the substitution \( u = -\frac{1}{x^2} \) with respect to \( x \). We find: \( \frac{du}{dx} = \frac{2}{x^3} \). Thus, \( du = \frac{2}{x^3} \, dx \), or \( dx = \frac{x^3}{2} \, du \).
04
Rewrite the Integral with Substitution
Replace \( dx \) and the corresponding parts of the integral in terms of \( u \):\[\int \frac{e^{-1 / x^2}}{x^3} \, dx = \int e^u \frac{x^3}{x^3} \cdot \frac{1}{2} \, du = \frac{1}{2} \int e^u \, du.\]
05
Integrate with Respect to the New Variable
The integral \( \frac{1}{2} \int e^u \, du \) is straightforward to solve. It results in:\[ \frac{1}{2} e^u + C, \]where \( C \) is the constant of integration.
06
Substitute Back to Original Variable
Substitute \( u = -\frac{1}{x^2} \) back into the expression to return to terms of \( x \):\[ \frac{1}{2} e^{-1 / x^2} + C. \]
07
Confirm and Simplify the Final Answer
The final answer, after substituting back, is: \[ \frac{1}{2} e^{-1 / x^2} + C. \] This confirms that the integral was properly evaluated using the substitution method.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integral Evaluation
When evaluating integrals, the main goal is to find the antiderivative of a given function. This process can sometimes be straightforward, but often, especially with complex functions, it requires different techniques.
The integral we are dealing with, \( \int \frac{e^{-1 / x^{2}}}{x^{3}} \, dx \), is not one that can be readily integrated with basic rules. Instead, we need to employ a more strategic approach.
The integral we are dealing with, \( \int \frac{e^{-1 / x^{2}}}{x^{3}} \, dx \), is not one that can be readily integrated with basic rules. Instead, we need to employ a more strategic approach.
- Identify the complexity of the integral, such as non-linear terms in the exponent or products of functions.
- Select an appropriate method, such as substitution or integration by parts, to simplify the integral.
- Perform the integration and simplify the result, if possible.
Exponential Functions
Exponential functions, like \( e^x \), are a special kind of function where the variable is in the exponent. These functions have unique properties that make them essential in mathematics, especially in integral calculus.
The function \( e^x \) has a distinctive characteristic: its derivative is itself, \( \frac{d}{dx}(e^x) = e^x \).
When dealing with integrals involving exponential functions, it's often useful to recognize this property.
The function \( e^x \) has a distinctive characteristic: its derivative is itself, \( \frac{d}{dx}(e^x) = e^x \).
When dealing with integrals involving exponential functions, it's often useful to recognize this property.
- The basic integral of an exponential function is straightforward, \( \int e^x \, dx = e^x + C \).
- More complex exponential functions inside integrals might require substitution to simplify the exponent.
- These functions frequently appear in physics, engineering, and natural sciences due to their growth and decay properties.
U-Substitution
U-substitution is a powerful technique for evaluating integrals, particularly when dealing with complicated expressions. It involves selecting a substitution that transforms the integral into a simpler form. Here's how it works:
This not only makes the integral manageable but also easier to evaluate.
- Choose a new variable \( u \) to represent a part of the function inside the integral, aiming to simplify it.
- Find the derivative of \( u \) with respect to \( x \), \( \frac{du}{dx} \), and express \( dx \) in terms of \( du \).
- Re-formulate the integral with \( u \) and complete the integration.
- Finally, substitute back the original variable into your result.
This not only makes the integral manageable but also easier to evaluate.
