Problem 18 - Thomas Calculus in SI Units

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

Step 1: Recognize the Integral Format

The integral given is \( \int (5 - 6x) \, dx \). This is a basic polynomial integral of the form \( \int (a + bx) \, dx \) where \( a = 5 \) and \( b = -6 \).

Step 2: Apply the Power Rule of Integration

To integrate a polynomial like \( 5 - 6x \), apply the power rule: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \). In this case, integrate each term separately: \[ \int 5 \, dx = 5x \]\[ \int -6x \, dx = -6 \cdot \frac{x^2}{2} = -3x^2 \]

Step 3: Write the General Antiderivative

Combine the integrated terms from Step 2: \[ 5x - 3x^2 + C \]where \( C \) is the constant of integration.

Step 4: Check by Differentiation

Differentiate the result from Step 3 to verify correctness: \[ \frac{d}{dx}(5x - 3x^2 + C) = 5 - 6x \]The derivative matches the original integrand \( 5 - 6x \), confirming that the antiderivative is correct.

Key Concepts

Understanding Indefinite Integrals

Indefinite integrals are a fundamental concept in calculus. They represent the "reverse" of differentiation and help us find the most general form of antiderivatives. Unlike definite integrals, which calculate the area under a curve within specific limits, indefinite integrals do not have set limits. As a result, the answer obtained is a function plus a constant, often denoted with a "\(C\)." This constant of integration represents the infinite number of antiderivatives a function can have.

When you see an integral symbol \(\int\) followed by a function and ending with \(dx\) or another differential, that typically signifies an indefinite integral. This concept enables us to find all possible functions whose derivative is the given integrand. The process involves finding a function such that when differentiated, it gives back the original function within the integral.

In summary, indefinite integrals are essential because they allow us to reverse the process of differentiation, offering a broad picture of families of functions that satisfy an initial function's derivative. This broad understanding is crucial for solving differential equations and analyzing physical systems.

Applying the Power Rule of Integration

The power rule of integration is a key tool used to integrate functions, especially polynomials. It states that to integrate \(x^n\), you increment the exponent by one and then divide by the new exponent, forming the integral \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) for \(n eq -1\). This rule simplifies the integration of polynomial functions, which consist of sums of terms involving powers of \(x\).

For example, given a function \(5 - 6x\), we treat each part separately using the power rule:

• The integral of a constant, like 5, is straightforward: \(\int 5 \, dx = 5x\).
• For the term \(-6x\), applying the power rule gives us \(-6 \cdot \frac{x^2}{2} = -3x^2\).

A key point to note is that the power rule does not apply when the exponent is \(-1\). Instead, for \(\frac{1}{x}\), the integral is the natural logarithm of \(x\), \(\ln |x| + C\). The power rule makes it easier to tackle both simple and complex polynomials, laying the groundwork for further integration techniques.

Polynomials and Their Integrals

Polynomials are expressions made up of terms involving variables raised to whole number powers. These terms are often summed with coefficients, like \(a + bx + cx^2\), making them algebraic expressions. In calculus, integrating polynomials is a frequent task, as polynomials are often used to model a variety of functions.

The integration of polynomials involves applying the power rule to each term in the polynomial separately. For example, given the polynomial \(5 - 6x\), you would integrate each part to find the antiderivative. As shown, the integral of 5 becomes \(5x\) (since 5 can be seen as \(5x^0\)), and \(-6x\) becomes \(-3x^2\) when integrated.

The result of integrating a polynomial is itself a polynomial (with higher-degree terms). Remember to always add the constant of integration, \(C\), because indefinite integrals signify a family of functions, not a single unique function. This constant is crucial when considering boundary conditions or initial values in problem-solving. Polynomials are central in calculus because they are simple to differentiate and integrate, making them useful approximations for more complex functions.