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Chapter 5: Problem 4

use the Exponential Rule to find the indefinite integral. $$ \int e^{-0.25 x} d x $$

Short Answer

Expert verified
The indefinite integral of \( e^{-0.25 x} \) is \( -4e^{-0.25 x} + C \).

Step by step solution

01

Identify the function to be integrated

The function to be integrated here is \( e^{-0.25 x} \).
02

Apply the Exponential Rule

Next, apply the exponential rule for integration: \( \int e^{kx} dx = 1/k e^{kx} + C \). In this case, \( k = -0.25 \). So, the integral becomes: \( \int e^{-0.25 x} dx = -4e^{-0.25 x} + C \).
03

Write the final result

The indefinite integral of \( e^{-0.25 x} \) is \( -4e^{-0.25 x} + C \).

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Key Concepts

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

Exponential Rule
The exponential rule is a powerful tool in calculus integration. It helps simplify the process of integrating exponential functions. The general form of the exponential rule is: \( \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \). Here, \( k \) is a constant, and \( C \) is the constant of integration. This rule makes integrating functions of the form \( e^{kx} \) much simpler by bypassing the need for more complex integration techniques. Instead, you only need to identify the constant \( k \), and then apply the formula.
For example, when you have a function like \( \int e^{-0.25x} \, dx \), you recognize \( k = -0.25 \), plug it into the formula, and compute the integral directly.
Indefinite Integral
Indefinite integrals represent a family of functions whose derivative results in the original function we started with. When working with indefinite integrals, we use the notation \( \int f(x) \, dx \), indicating we are trying to find the function whose derivative is \( f(x) \). Importantly, indefinite integrals always include an arbitrary constant, \( C \). This is because the derivative of a constant is zero, meaning there is no way to tell if there was a constant term in the original function. Thus, the result of an indefinite integral, such as \( \int e^{-0.25x} \, dx = -4e^{-0.25x} + C \), includes this constant to account for any potential original constants.
Integration Techniques
Integration techniques refer to a variety of methods used to find the integral of functions. These techniques help tackle a wide range of functions that might not be straightforward to integrate. Here are some common techniques:
  • Substitution Method: Useful when an integral contains a composite function. It simplifies the integral into a more manageable form.
  • Integration by Parts: Handy for functions that are products of simpler functions. It's applicable when substitution isn't straightforward.
  • Partial Fractions: This method breaks down complex rational expressions into simpler ones that are easier to integrate.
  • Trigonometric Integrals: Used specifically for integrals involving trigonometric functions, utilizing identities to simplify expressions.
In our case with \( \int e^{-0.25x} \, dx \), we used the exponential rule, which is a specific integration technique suited for exponential functions. Understanding when to use these different techniques can vastly simplify solving integrals.

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Most popular questions from this chapter

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