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Chapter 13: Problem 10

Evaluate the integrals. $$ \int(4-x) d x $$

Short Answer

Expert verified
The short answer to the integral of the function \( (4-x) \) with respect to x is: \( \int(4 - x) dx = 4x - \frac{1}{2}x^2 + C \), where C is the constant of integration.

Step by step solution

01

Rewrite the function for clarity

We can rewrite the function (4-x) as (4 - 1x) to make it easier to see the coefficients and exponents that we'll be working with. So, we have: \( \int(4 - x) dx = \int(4 - 1x) dx \)
02

Integrate each term separately

Now, we integrate each term in the expression separately with respect to x: For the constant term 4, the integral is: \( \int 4 dx = 4x \) For the term -1x, the integral is (applying the power rule for integration): \( \int (-1x) dx = \frac{-1x^{(1+1)}}{(1+1)} = -\frac{1}{2}x^2 \)
03

Combine the results

Now that we've integrated each term separately, let's combine them to get the final integrated function: \( \int(4 - x) dx = 4x - \frac{1}{2}x^2 + C \) Here, C is the constant of integration, which could be any constant value. So, the final solution for the integral is: \( \int(4 - x) dx = 4x - \frac{1}{2}x^2 + C \)

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Key Concepts

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

Integral Calculus
Integral calculus is a branch of mathematics focused on the concept of integration. It deals with the accumulation of quantities and the areas under and between curves. This discipline is essentially about 'summing up' an infinite number of infinitesimally small quantities. Here, the main object is the integral, which can be definite or indefinite.
  • **Definite Integral**: Provides the area under a curve between two points.
  • **Indefinite Integral**: Gives a general form of antiderivatives, accompanied by a constant of integration.

In the context of the problem, we are seeking the indefinite integral of \( 4-x \), which means finding the antiderivative without specific bounds.
Power Rule for Integration
The power rule for integration is a key tool for simplifying the process of finding integrals. It provides a straightforward way to integrate functions of the form \( x^n \), where \( n eq -1 \). The rule states:
\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]
Here, **n** represents the exponent of the variable \( x \), and **C** is the constant of integration, added to every indefinite integral result.
For example, using this rule in our expression \( -1x \), we see it as \( -1x^1 \). Applying the power rule, the integral becomes \[-\frac{1}{2}x^2\], which gives us the new exponent and coefficient.
It's a simple yet powerful technique to find antiderivatives without much hassle.
Constant of Integration
In indefinite integration, the constant of integration, represented as **C**, is crucial because it encompasses the infinite family of possible antiderivatives. When we integrate, we are essentially reversing differentiation. However, differentiation 'loses' the constant due to its derivative being zero.
  • **Purpose**: The constant **C** ensures our antiderivative formula covers all potential functions, each differing by some constant.
  • **Significance**: In practical problems, the value of **C** can be determined if extra conditions or initial values are provided.
In our problem, after integrating the terms and obtaining \( 4x - \frac{1}{2}x^2 \), we add **C** to indicate that any constant can be part of the antiderivative solution. This signifies all possible vertical shifts of the function on a graph.

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