Problem 46 - Applied Calculus

Question

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{e^{1 / x}}{x^{2}} d x \quad\left[\text { Hint: Let } u=\frac{1}{x} .\right] $$

Step-by-Step Solution

Answer

\( \int \frac{e^{1/x}}{x^2} \, dx = -e^{1/x} + C \).

Step 1: Substitute with 'u'

We start by substituting the expression inside the integral by letting \( u = \frac{1}{x} \). This substitution suggests that we need to express \( dx \) in terms of \( du \).

Step 2: Differentiate 'u'

Differentiate \( u = \frac{1}{x} \) with respect to \( x \) to find \( dx \) in terms of \( du \). We find that \( du = -\frac{1}{x^2} \, dx \). Thus, \( dx = -x^2 \, du \).

Step 3: Substitute and Simplify the Integral

Substitute \( u \) and \( dx \) into the integral: \[ \int \frac{e^{1/x}}{x^2} \, dx = \int e^u (-x^2) \, du \]. Since \( \frac{1}{x} = u \), \( x^2 = \frac{1}{u^2} \). Substitute this into the expression to get: \[ -\int e^u \, du \].

Step 4: Solve the Integral

Integrate \( -\int e^u \, du \), which results in \( -e^u + C \), where \( C \) is the constant of integration.

Step 5: Back-substitute 'u'

Back-substitute \( u = \frac{1}{x} \) to express the result in terms of \( x \). The solution to the integral is \( -e^{1/x} + C \).

Key Concepts

Indefinite Integrals

An indefinite integral is an essential concept in calculus. It represents the most general form of an antiderivative, which is basically the reverse process of differentiation.
When you perform indefinite integration, you're essentially finding a function whose derivative yields the original function inside the integral. This process does not have specific limits of integration, hence the term "indefinite."Key points about indefinite integrals include:

• The notation used is \( \int f(x) \, dx \), where \( f(x) \) is the function to be integrated.
• The result is a family of functions, expressed as \( F(x) + C \), where \( C \) is an arbitrary constant called the constant of integration.
• Indefinite integrals lack specific boundary values, distinct from definite integrals which do have limits.

This ambiguity from the constant \( C \) accounts for any constants that vanish during differentiation. As seen in the provided exercise, when substitutions and simplifications are complete, you always add \( C \) to denote all possible antiderivatives.

Integration Techniques

Integration techniques are strategies applied to solve more complex integrals. Different techniques suit different forms and structures of functions. The substitution method is especially valuable when functions contain compositions, such as nested or chain-function forms.
For example, if you encounter a function with a composition, like \( e^{1/x} \), using substitution simplifies the problem. Often, choosing an inner function of a composite as a substitute simplifies the integral greatly. Here's the process:

• Select the substitution: In our exercise, letting \( u = \frac{1}{x} \) is logical.
• Find the derivative: Differentiate \( u \) to express \( dx \) in terms of \( du \).
• Substitute: Replace all parts of the integral, including \( dx \), with terms of \( u \) and \( du \).
• Integrate: Now integral in terms of \( u \) simplifies and can be solved easily, such as \( -\int e^u \, du \).
• Back-Substitute: Replace \( u \) with the original terms to express the result in terms of the original variable \( x \).

This method won't work for every integral but is especially effective for nested functions or products of a function and its derivative.

Change of Variables

The change of variables is an approach that transforms the variables of an integral into a more manageable form. This is particularly useful when the original variables lead to complicated integrals.
In an integral, a change of variables typically involves:

• Choosing a new variable that simplifies the integral (new variable substitution).
• Expressing all parts of the integral, both the function and the differential \( dx \), in terms of this new variable.
• This new variable, say \( u \), is often selected because it appears frequently in the function to be integrated.

A change of variables is akin to re-centering the problem, allowing problematic parts of the integral to become more straightforward.
In the exercise provided, changing the variable to \( u = \frac{1}{x} \), allowed translating the challenge presented by the exponential function into a simplified integral of \( e^u \), which is easily solvable. This tactic is common when function derivatives are present, as it exploits the simplification pattern available within different functions' relationships.