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Chapter 4: Problem 4

Determine these indefinite integrals. $$\int 4 \, d x$$

Short Answer

Expert verified
4x + C

Step by step solution

01

Understand the Integral

Recognize that the integral is given in the form \( \int c \, dx \) where \( c \) is a constant. In this case, \( c = 4 \).
02

Apply the Power Rule for Integration

The power rule for integration says that \int c dx\ = cx + C, where \( c \) is a constant and \( C \) is the constant of integration.
03

Integrate the Constant 4

Using the power rule, integrate \int 4 dx\: It becomes \( 4x + C \), where \( C \) is the constant of integration.

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Key Concepts

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

integration
Integration is a fundamental concept in calculus. It combines infinitely many small data points to find the total. Think of it as the opposite of differentiation, which breaks things down into small parts. The process of integration helps us calculate areas under curves, among other things. It is represented by the symbol \( \int ... dx \).
In the exercise, you are working with an indefinite integral, which means there is no specific range of values to integrate over. So, the result includes an added constant, called the constant of integration.
power rule
The power rule for integration is a simple but crucial tool. It helps us integrate functions of the form \( x^n \).
The power rule states that for any constant \( n eq -1 \),
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] where \( C \) is the constant of integration.
In the given exercise, although we're integrating a constant 4, the power rule still applies. We treat the constant like \( x^0 \).
Thus, \[ \int 4 \, dx = 4x + C \] Integrating constants is straightforward with the power rule.
constant of integration
Every indefinite integral includes a constant of integration, denoted by \( C \).
This arises because the process of differentiation makes any constant disappear (e.g., \( d/dx (4x + 7) = 4 \)). So when integrating, we re-introduce this constant to account for all possible original functions.
In the exercise, after integrating, you get \( 4x \). To represent the general solution, add \( C \), giving \( 4x + C \).
This tells us any function of the form \( 4x + C \) could be the original function before differentiation.

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