Problem 21 - Thomas Calculus

Question

In Exercises \(17-56,\) 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. $$\int\left(2 x^{3}-5 x+7\right) d x$$

Step-by-Step Solution

Answer

The most general antiderivative is \( \frac{1}{2}x^4 - \frac{5}{2}x^2 + 7x + C \).

Step 1: Identify the Integral Components

The integral to solve is \( \int (2x^3 - 5x + 7) \ dx \). It consists of three separate terms: \( 2x^3 \), \( -5x \), and \( 7 \). We'll find the antiderivative of each term separately.

Step 2: Integrate Each Term Separately

Integrate the polynomial terms one at a time: 1. The antiderivative of \( 2x^3 \) is \( \frac{2}{4}x^4 = \frac{1}{2}x^4 \).2. The antiderivative of \( -5x \) is \( -\frac{5}{2}x^2 \).3. The antiderivative of the constant \( 7 \) is \( 7x \).

Step 3: Combine Antiderivatives

Combine the antiderivatives from each term:\[ \int (2x^3 - 5x + 7) \ dx = \frac{1}{2}x^4 - \frac{5}{2}x^2 + 7x + C \]where \( C \) is the arbitrary constant of integration.

Step 4: Verify by Differentiation

Differentiate the resulting expression \( \frac{1}{2}x^4 - \frac{5}{2}x^2 + 7x + C \): 1. Differentiate \( \frac{1}{2}x^4 \) to get \( 2x^3 \).2. Differentiate \( -\frac{5}{2}x^2 \) to get \( -5x \).3. Differentiate \( 7x \) to get \( 7 \).Combine the terms to confirm: \( 2x^3 - 5x + 7 \), which matches the original integrand, verifying the solution.

Key Concepts

Understanding Antiderivatives

When we talk about antiderivatives, we're dealing with the reverse process of differentiation. This means finding a function whose derivative gives us the original function. So, in simple terms, an antiderivative of a function \( f(x) \) is another function \( F(x) \) such that \( F'(x) = f(x) \). In integration, the goal is to find this \( F(x) \).

In our exercise, the antiderivative process was used to determine the function form from \( 2x^3 - 5x + 7 \), ultimately leading to the expression \( \frac{1}{2}x^4 - \frac{5}{2}x^2 + 7x + C \). Notice the presence of different powers of \( x \). Each component will be handled individually to find their respective antiderivatives.

The Role of Polynomial Integration

Polynomial integration involves working with polynomial expressions and finding their antiderivatives. This process is typically straightforward because polynomials have predictable derivatives.

Here’s how we manage polynomial integration:

• Identify each term in the polynomial separately.
• Apply the power rule to compute the antiderivative, which states: if the term is \( ax^n \), its integral will be \( \frac{a}{n+1}x^{n+1} \).

In the provided exercise, we integrated terms like \( 2x^3 \), \( -5x \), and \( 7 \) separately to get their antiderivatives, \( \frac{1}{2}x^4 \), \( -\frac{5}{2}x^2 \), and \( 7x \), respectively. Polynomial terms integrate to new forms with increased exponents, reflecting the gradual accumulation of quantity over a range.

Importance of the Constant of Integration

The constant of integration, denoted by \( C \), represents any constant value that could have been lost during the differentiation process. This is because when you differentiate a constant, it becomes zero, making it seem like it was never there initially.

Therefore, in indefinite integration, it is crucial to include \( C \) to reflect all possible antiderivatives of a function. In practical situations, this constant offers a full spectrum of solutions, from which a particular one can be selected based on specific initial conditions or constraints.

For instance, integrating the polynomial in our exercise, \( 2x^3 - 5x + 7 \), results in \( \frac{1}{2}x^4 - \frac{5}{2}x^2 + 7x + C \). Without \( C \), we limit its applicability in real-world problems that might need specific function values.