Problem 14 - Applied Calculus

Question

Find each indefinite integral. \(\int(2-4 x) d x\)

Step-by-Step Solution

Answer

The indefinite integral is \( 2x - 2x^2 + C \).

Step 1: Identify the terms to integrate

The integral given is \( \int (2 - 4x) \, dx \). Notice that there are two terms: a constant \(2\) and a linear term \(-4x\). We will integrate each of these terms separately.

Step 2: Integrate each term separately

First, integrate the constant term \(2\). The integral of a constant \(c\) with respect to \(x\) is \(cx\). Therefore, \( \int 2 \, dx = 2x \).Next, integrate the linear term \(-4x\). Using the power rule for integration, \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \), where \(n\) is 1. Thus, \( \int -4x \, dx = -4 \cdot \frac{x^2}{2} = -2x^2 \).

Step 3: Combine the results

Now combine the results from integrating each term. \( \int (2 - 4x) \, dx = 2x - 2x^2 \).

Step 4: Add the constant of integration

Since this is an indefinite integral, we must add the constant of integration \(C\). Therefore, the final answer is \( 2x - 2x^2 + C \).

Key Concepts

Integration

Integration is a fundamental concept in calculus that deals with finding an antiderivative or the area under a curve. In simple terms, it is the reverse process of differentiation. If differentiation helps us find the rate at which something changes, integration helps us find the total accumulation of that change.

When performing integration, especially indefinite integration, we aim to determine the function that, when derived, leads us back to the original expression we started with. In this process, no specific limits or boundaries are defined, so we end up with a general form of the solution rather than a numeric result.

• Definite vs. Indefinite Integrals: Indefinite integrals do not have set limits and include a constant of integration, whereas definite integrals evaluate the function between two specific bounds to yield a number.
• Basic Symbols: The integral sign \( \int \) is used along with \( dx \) to denote integration concerning the variable \( x \).

This can be very useful in physics and engineering, where building a broad understanding is more valuable than a specific point calculation.

Power Rule

The power rule is a straightforward technique used when integrating polynomials. Just as in differentiation, where you multiply and drop the power, here we add one to the exponent and divide by the new exponent.

Suppose we have an expression like \( x^n \). To integrate it using the power rule, follow these steps:

• Add one to the exponent: For \( x^n \), it becomes \( x^{n+1} \).
• Divide by the new exponent: The integral becomes \( \frac{x^{n+1}}{n+1} \).

This rule comes in handy when dealing with terms like constants or monomials. For example, in the case of \( -4x \), applying the power rule gives \( -4 \cdot \frac{x^2}{2} = -2x^2 \).

Remember, the power rule works nicely for polynomial terms, but you need other methods, like integration by parts or substitution, for more complex forms involving trigonometric, exponential, or logarithmic functions.

Constant of Integration

When we deal with indefinite integrals, the "constant of integration" is an indispensable element to understand. Whenever we integrate a function, there could be a family of functions as solutions. These functions differ by a constant, since the derivative of any constant is zero.

The constant of integration \( C \) symbolizes this family by telling us that there might be multiple functions which derivative could lead to the same result. For instance, integrating \( \int (2 - 4x) dx \) gives \( 2x - 2x^2 + C \). The \( C \) is crucial as it accounts for any vertical shift that might be present in the antiderivative function.

• Why add \( C \)? It preserves the generality of solutions and is vital in solving, particularly in differential equations where initial conditions later determine its value.
• Not to be missed: Failing to include \( C \) means missing possible solutions, potentially leading to errors in calculations involving broader mathematical concepts.

Understanding the constant of integration is fundamental in ensuring the completeness of indefinite integrals and addressing potential solution variations.