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Chapter 5: Problem 3

Evaluate. $$ \int 100 d x $$

Short Answer

Expert verified
The integral evaluates to \( 100x + C \).

Step by step solution

01

Understand the Integral

We need to evaluate the indefinite integral \( \int 100 \, dx \). This integral involves a constant function, 100, which means we will find the antiderivative of a constant function.
02

Apply the Power Rule for Integration

For a constant \( c \), the integral \( \int c \, dx \) is simply \( cx + C \), where \( C \) is the constant of integration. In this case, \( c = 100 \).
03

Compute the Indefinite Integral

Using the information from the previous step, we integrate \( 100 \). Therefore, \( \int 100 \, dx = 100x + C \).

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Key Concepts

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

Antiderivative
When we talk about finding an antiderivative, we're discussing the process of determining a function whose derivative is the given function. Simply put, if you take the derivative of the antiderivative, you get back the original function. This is a core concept in calculus and is fundamental for solving indefinite integrals.

In the context of the exercise, the goal is to find an antiderivative of the constant function 100. Imagine a function whose slope is always zero, such as a horizontal line at 100. The antiderivative of this function would give us a new function that increases linearly with 100. Thus, the expression to solve is a search for this antiderivative.
Power Rule for Integration
The Power Rule for Integration is one of the most crucial techniques for solving integrals. It states that the integral of a power function can be found by increasing the exponent by one and dividing by the same number.

Mathematically, for any real number \( n eq -1 \), the integral of \( x^n \) is \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \].
This rule can also be applied to constant functions like in our exercise. Although there is no variable \( x \) in \( 100 \), you can think of it as \( 100x^0 \). Integrating gives \( \frac{100x^{0+1}}{0+1} = 100x \). This simplifies our task, allowing us to integrate constant functions quickly.
Constant of Integration
Every indefinite integral contains a constant of integration, often represented by \( C \). This constant is pivotal because it accounts for all the possible vertical shifts of the antiderivative.

Why do we need this constant? Well, in differentiation, many functions could have the same derivative but differ by a constant value. The integral must account for this ambiguity, introducing \( C \) to represent any constant number that was lost during differentiation.
  • It ensures our solutions cover all potential antiderivatives.
  • In our example, after integrating 100, we write \( 100x + C \) to encompass all possible functions whose derivative is 100.
Always remember to add \( C \) when solving indefinite integrals to ensure completeness.

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