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Chapter 5: Problem 3

use the Exponential Rule to find the indefinite integral. $$ \int e^{4 x} d x $$

Short Answer

Expert verified
The indefinite integral of \( e^{4x} \) with respect to \( x \) is \( (1/4) * e^{4x} + C \)

Step by step solution

01

Understand the exponential rule

The exponential rule for integration states that the integral of \( e^{ax} \) with respect to \( x \) is \( (1/a) * e^{ax} + C \). Here, \( C \) is the constant of integration.
02

Identify a and apply the rule

From, \( e^{4x} \), we can see that \( a = 4 \). Therefore, applying the exponential rule, we have: \( (1/4) * e^{4x} + C \).
03

Write out the final answer

Upon writing out the expression from Step 2, we get the final answer: \( (1/4) * e^{4x} + C \).

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Key Concepts

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

Indefinite Integral
In calculus, the concept of an indefinite integral is central to understanding how to reverse the process of differentiation. When we talk about an indefinite integral, we are looking for a function whose derivative is the given function. It represents a family of functions, given the differentiation rules.
For example, the indefinite integral of a function \( f(x) \) is denoted as \( \int f(x)\,dx \). This process helps in determining functions known as "antiderivatives".
  • An important aspect of indefinite integrals is that they include an arbitrary constant \( C \), because taking the derivative of a constant is zero and it gets "lost" during differentiation.
  • Without the + \( C \), we would not account for all possible antiderivatives of a function.
Let's consider an example: the indefinite integral of \( e^{4x} \). Applying the exponential rule, we find that it is \( (1/4) e^{4x} + C \), where \( C \) is any constant value.
Integration Techniques
When it comes to integration, choosing the right technique is essential to solving integrals effectively. There are different methods available, each suitable for certain types of functions.
Some of the most common integration techniques include:
  • Basic Integration Rules: Simple rules like \( \int k\, dx = kx + C \) and \( \int x^n\, dx = \frac{x^{n+1}}{n+1} + C \).
  • Substitution: Useful for functions that resemble derivatives of a composition of functions.
  • Integration by Parts: Applies to products of functions, often using the formula \( \int u\, dv = uv - \int v\, du \).
  • Trigonometric Integrals: Useful for integrating functions involving trigonometric expressions.
In our scenario, we employ the Exponential Rule. This rule is specially catered to integrate functions of the form \( e^{ax} \), like in the problem \( \int e^{4x} dx \). By recognizing \( a = 4 \), the resulting integration follows directly, simplifying our work considerably.
Constant of Integration
A unique component that emerges in the context of indefinite integration is the constant of integration, denoted as \( C \). This constant represents the family of antiderivatives and is critical for reflecting the generality of solutions in indefinite integrals.
  • When solving integrals, if we omit \( C \), our solution would inaccurately reflect all possibilities of the antiderivative function.
  • The presence of \( C \) ensures that any solution derived is "complete" and accommodating of any shifts or displacements existed.
For instance, in the indefinite integral of \( e^{4x} \) which results in \( (1/4)e^{4x} + C \), the \( C \) guarantees the solution is inclusive of all constant differences that could alter the antiderivatives yet still differentiate to the original function.

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Most popular questions from this chapter

Use a symbolic integration utility to evaluate the definite integral. \(\int_{2}^{5}\left(\frac{1}{x^{2}}-\frac{1}{x^{3}}\right) d x\)

Use the values \(\int_{0}^{5} f(x) d x=6\) and \(\int_{0}^{5} g(x) d x=2\) to evaluate the definite integral. (a) \(\int_{0}^{5} 2 g(x) d x\) (b) \(\int_{5}^{0} f(x) d x\) (c) \(\int_{5}^{5} f(x) d x\) (d) \(\int_{0}^{5}[f(x)-f(x)] d x\)

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