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Chapter 7: Problem 19

Evaluate the integrals. $$\int \frac{e^{1 / x}}{x^{2}} d x$$

Short Answer

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

Step by step solution

01

Identify the Suitable Substitution

The integral is \( \int \frac{e^{1/x}}{x^2} \, dx \). Start by identifying a substitution that simplifies the integrand. Notice the presence of \( \frac{1}{x} \) inside the exponent and \( x^{-2} \) in front. Let \( u = \frac{1}{x} \), which leads to \( du = -\frac{1}{x^2} \, dx \). Therefore, \( dx = -x^2 \, du \).
02

Substitute into the Integral

Replace \( \frac{1}{x} \) with \( u \) and \( dx \) with \( -x^2 \, du \) from the substitution \( u = \frac{1}{x} \). The integral becomes:\[ \int e^u (-du) = - \int e^u \, du \]
03

Integrate with Respect to the New Variable

The integral \( -\int e^u \, du \) is a straightforward integration. Since the antiderivative of \( e^u \) is \( e^u \), we have:\[ -e^u + C \] where \( C \) is the constant of integration.
04

Substitute Back to the Original Variable

Substitute \( u = \frac{1}{x} \) back into the antiderivative:\[ -e^{\frac{1}{x}} + C \] This returns the expression to terms of the original variable \( x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration by Substitution
Integration by substitution is a technique that helps simplify complex integrals by changing variables. It's akin to the reverse of the chain rule in differentiation. In our exercise, this technique was used to tackle the integral \( \int \frac{e^{1/x}}{x^2} \, dx \).
  • First, we identify a substitution that simplifies the integrand. In this case, we set \( u = \frac{1}{x} \).
  • This substitution leads to \( du = -\frac{1}{x^2} \, dx \), so \( dx \) can be expressed as \(-x^2 \, du \).
  • By substituting back, the integral becomes \(-\int e^u \, du \).
Using this method, we transformed a complicated integral into a simpler one, making it easier to find the solution. Integration by substitution is a powerful tool when dealing with integrals of functions where direct integration is not feasible.
Exponential Functions
Exponential functions appear frequently in calculus, characterized by the constant base \( e \), Euler's number, raised to some power. In the given exercise, the integrand had an exponential function \( e^{1/x} \). Understanding exponential functions is crucial because:
  • The derivative and the antiderivative of \( e^x \) are both \( e^x \), making calculus operations particularly straightforward.
  • Substitutions involving exponential functions often simplify the integrals considerably, as seen in this problem.
The nature of exponential growth and decay is modeled by these functions, making them essential in various applications, from natural sciences to economics. Recognizing when to use and how to manipulate them can significantly ease solving complex integrals.
Antiderivative
Finding the antiderivative, or the indefinite integral, is the process of reversing differentiation. In our exercise, once the integral was rewritten through substitution, it involved finding the antiderivative of \( e^u \). Here are some insights:
  • The antiderivative of \( e^u \) is straightforwardly \( e^u \), which makes it unique among functions.
  • To complete the integration, we multiply by the transformation factor from substitution, which in this case was \(-1\).
Thus, the antiderivative becomes \(-e^u + C\). This part of solving integrals always brings us closer to understanding the original function's behavior over an interval.
Constant of Integration
The constant of integration, denoted as \( C \), emerges when calculating indefinite integrals. After integrating \( -\int e^u \, du \), we obtained \( -e^u + C \). Why is this constant important?
  • Any indefinite integral has infinitely many solutions because the derivative of a constant is zero. This means we add \( C \) to represent all such possibilities.
  • This constant ensures there's no loss of generality and is crucial when solving definite integrals as it cancels out appropriately.
By considering the constant of integration, we leave room to encapsulate all possible vertical shifts of the antiderivative graph, providing a complete set of solutions and honoring the functions involved.

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Most popular questions from this chapter

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\left(\theta^{2}+2 \theta\right) \tanh ^{-1}(\theta+1)$$

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