Problem 46 - Thomas Calculus

Question

Evaluate the integrals. \begin{equation}\int \frac{e^{-1 / x^{2}}}{x^{3}} d x\end{equation}

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{2} e^{-1 / x^2} + C \).

Step 1: 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.

Step 2: 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.

Step 3: 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 \( dx = \frac{x^3}{2} \, du \).

Step 4: Rewrite the Integral with Substitution

Replace \( dx \) and the corresponding parts of the integral in terms of \( u \):\[\int \frac{e^{-1 / x^2}}{x^3} \, dx = \int e^u \frac{x^3}{x^3} \cdot \frac{1}{2} \, du = \frac{1}{2} \int e^u \, du.\]

Step 5: 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.

Step 6: Substitute Back to Original Variable

Substitute \( u = -\frac{1}{x^2} \) back into the expression to return to terms of \( x \):\[ \frac{1}{2} e^{-1 / x^2} + C. \]

Step 7: 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.

Key Concepts

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.

• 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.

Using a combination of these steps allows for the successful evaluation of integrals that might initially seem difficult.

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 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.

In our original problem, the function \( e^{-1/x^2} \) involves an exponential function with a more complicated exponent, prompting the need for substitution.

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:

• 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.

In our problem, setting \( u = -\frac{1}{x^2} \) allows us to simplify the troublesome exponent. The derivative, \( \frac{du}{dx} = \frac{2}{x^3} \), matches well with the original integral's denominator, facilitating the substitution process.
This not only makes the integral manageable but also easier to evaluate.