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Chapter 6: Problem 9

Determine the following: $$\int 7 d x$$

Short Answer

Expert verified
\(\bigint{} 7 dx = 7x + C\)

Step by step solution

01

Identify the Integral

Recognize that the problem requires finding the integral of a constant function, which in this case is 7. The integral to solve is \(\frac{d}{dx}\big(7\big)dx\).
02

Apply the Constant Rule of Integration

Use the constant rule of integration, which states that the integral of a constant \(a\) with respect to \(x\) is given by \(ax + C\), where \(C\) is the constant of integration. For \(7\), this rule applies directly.
03

Perform the Integration

Integrate the constant 7 with respect to \(dx\). By applying the constant rule, we have: \(\frac{\bigint{} 7 dx}{=\big(7x + C\big)}\).
04

Include the Constant of Integration

Remember to add the integration constant \(C\) to account for any constant that could have been present before differentiation: \(7x + C\).

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Key Concepts

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

Integral of a Constant
When you see a problem asking to integrate a constant, like in the exercise \(\bigint{} 7 \, dx\), it is simpler than it may seem. A constant is a number that doesn't change, no matter what happens to the variable next to it. In mathematics, integrating a constant means finding the area under a horizontal line, which is quite straightforward. For example, a line at \(y = 7\) is always 7 units above the x-axis, no matter the value of \(x\). Hence, you are trying to sum many slices of height 7 over the width of \(dx\).
Constant Rule of Integration
The constant rule of integration is one of the simplest and most fundamental integration rules. It states that the integral of a constant \(a\) with respect to \(x\) is given by \(ax + C\). Here, \(C\) represents an unknown constant. So, if you are faced with an integral such as \(\bigint{}a\,dx\), you can directly write it as \(\big( ax + C\big)\).

For example, in our exercise with \(7\) as the constant, applying the constant rule of integration means the integral can be written straightaway as \(7x + C\).
Integration Constant
After performing the integration of a function, it is essential to add an integration constant \(C\). Why? Because in calculus, when you differentiate a constant term, it becomes zero. This means that any constant could have been there before differentiation. Therefore, while integrating, we add \(C\) at the end to represent this unknown constant.
  • For \(\bigint{} 7\, dx\), after integrating, we get \(\big(7x + C\big)\).
  • This ensures that our solution represents all possible constants that could have originally been in the function.

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