Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x^{100}+x^{99} $$

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = \frac{x^{101}}{101} + \frac{x^{100}}{100} + C \).

Step 1: Identify the Function to Integrate

We need to find the antiderivative of the function \( f(x) = x^{100} + x^{99} \). This means finding a function \( F(x) \) such that \( F'(x) = f(x) \).

Step 2: Apply Basic Rules of Integration

Each term of the polynomial \( x^n \) integrates to \( \frac{x^{n+1}}{n+1} \). Apply this rule to each term separately.

Step 3: Integrate Each Term Separately

Integrate the first term: \[ \int x^{100} \, dx = \frac{x^{101}}{101} \] Integrate the second term: \[ \int x^{99} \, dx = \frac{x^{100}}{100} \]

Step 4: Combine the Integrated Terms

Add the results of the two integrals from Step 3 together: \[ F(x) = \frac{x^{101}}{101} + \frac{x^{100}}{100} + C \] where \( C \) is the constant of integration.

Key Concepts

Integration Rules

Integration is essentially the reverse process of differentiation. When we integrate, we're looking for a function whose derivative will bring us back to the original function we started with. The key rule for integration is the Power Rule, which is used frequently in calculus.

This rule states: If you have a term like \( x^n \), you can integrate it to \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. This constant is necessary because the process of differentiation of a constant gives zero; therefore, it disappears during differentiation and reappears in integration.

• Power Rule: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)
• Constant Rule: Integrating a constant \( c \) gives \( cx + C \)
• Sum Rule: The integral of a sum is the sum of integrals \( \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \)
• Difference Rule: The integral of a difference is the difference of integrals \( \int (f(x) - g(x)) \, dx = \int f(x) \, dx - \int g(x) \, dx \)

Understanding these foundational rules helps us tackle even the most complex integration tasks. Each function's integral will combine these rules in unique and interesting ways.

Polynomial Integration

Polynomial functions are sums of terms consisting of a variable raised to a non-negative integer power. These are among the simplest functions to integrate because they directly employ the Power Rule.

When dealing with polynomials, we break down the polynomial into individual terms, integrate each term separately using the Power Rule, and then recombine the integrated terms. This process is quite systematic, making polynomial integration straightforward.

• Take each term of a polynomial separately, e.g., \( x^{100} \) and \( x^{99} \).
• Apply the Power Rule: adjust the exponent correctly and divide by the new exponent.
• Combine your results to find the general antiderivative: for instance, \( \frac{x^{101}}{101} + \frac{x^{100}}{100} + C \).

This method is reliable and consistent. It ensures that you can easily find the antiderivative of any polynomial function, reinforcing the simple beauty of calculus.

Calculus Concepts

Calculus is the mathematical study of change, and it comprises several core concepts. Integration is one of these, serving as a fundamental operation in calculus. It is used to find areas under curves, solve differential equations, and evaluate accumulated quantities.

Understanding calculus concepts such as derivatives and integrals is essential for grasping how they interrelate. They are inverse operations:

• Differentiation: Finding the rate at which a function changes.
• Integration: Finding a function given its rate of change; it's like going backwards from a derivative to find the original function.
• Antiderivative: The function you get from integrating, plus the constant \( C \), which represents any possible constant that when differentiated, will zero out.

Understanding these basics equips you with the skills to apply calculus to real-world problems, like motion, growth, and change in various fields.