Problem 26 - University Calculus: Early Transcendentals

Question

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int(5-6 x) d x$$

Step-by-Step Solution

Answer

The most general antiderivative is \(5x - 3x^2 + C\).

Step 1: Identify Each Term

The given integral is \( \int(5 - 6x) \, dx \). This expression is composed of two terms: a constant \(5\) and a linear term \(-6x\).

Step 2: Integrate Each Term Separately

Using the linearity of the integral, integrate each term separately: 1. The integral of the constant \(5\) with respect to \(x\) is \(5x\).2. The integral of \(-6x\) is achieved by applying the power rule, producing \(-\frac{6}{2}x^2\) which simplifies to \(-3x^2\).

Step 3: Combine the Antiderivatives

Combine the antiderivatives of each term to get the general antiderivative: \[5x - 3x^2 + C\]where \(C\) is the constant of integration.

Step 4: Differentiate to Check the Solution

Differentiate the antiderivative \(5x - 3x^2 + C\):1. The derivative of \(5x\) is \(5\).2. The derivative of \(-3x^2\) is \(-6x\).3. The derivative of the constant \(C\) is \(0\).This differentiation confirms that the derivative is \(5 - 6x\), which matches the integrand.

Key Concepts

Indefinite Integrals

An indefinite integral, also known as an antiderivative, is a fundamental concept in calculus. When we calculate an indefinite integral, we essentially find the function whose derivative gives the original function. This process of finding the antiderivative helps us understand how a function accumulates change.
Unlike definite integrals, indefinite integrals do not have bounds. This is why they represent a family of functions rather than a single number. To express this family of functions, we include a constant of integration, denoted as \( C \).
So, when you integrate a function, you get an expression like \( \int f(x) \, dx = F(x) + C \), where \( F(x) \) is any antiderivative of \( f(x) \). The constant \( C \) accounts for the fact that multiple functions can have the same derivative, differing only by a constant.It is crucial to include \( C \) in the final answer to ensure the solution represents all possible antiderivatives of the integrand. Always remember to add this constant when you work with indefinite integrals.

Linearity of Integrals

The linearity of integrals is a handy property that simplifies the integration process. This principle states that the integral of a sum is equal to the sum of the integrals. Mathematically, if you have two functions, \( f(x) \) and \( g(x) \), then:
\[ \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \]
This property also applies to constants, allowing us to factor them out of the integral. For instance, \( \int a \cdot f(x) \, dx = a \cdot \int f(x) \, dx \).

• It helps in breaking down complex integrals into simpler parts that are easier to handle.
• When dealing with polynomials or sums of functions, you can integrate each term individually and then combine them.

Through the linearity of integrals, solving an integral becomes more manageable. You can separate each piece, deal with them individually, and finally, add them together to find the complete solution.

Power Rule in Integration

The power rule in integration is a straightforward technique used to integrate expressions where the variable is raised to a power. This rule resembles the power rule in differentiation but works in reverse.
If you have a function of the form \( x^n \), where \( n eq -1 \), the power rule states:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

The "+ C" is crucial, as it accounts for any constant that was differentiated away in the original function calculation.

Here’s how the power rule simplifies integration:

• Add one to the exponent \( n \).
• Divide the term by this new exponent \( n+1 \).
• Don't forget to add the constant of integration \( C \).

This rule is particularly useful for integrating polynomial functions or any terms that neatly fit the form \( x^n \). Thanks to the power rule, such tasks can be quickly computed with accuracy. Just be mindful of the exception; it doesn’t apply when \( n = -1 \), as this is a different case with its own rule.