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Chapter 25: Problem 25

Find antiderivatives of the given functions. $$f(x)=50 x^{99}-39 x^{-79}$$

Short Answer

Expert verified
The antiderivative of \( f(x) = 50x^{99} - 39x^{-79} \) is \( F(x) = 0.5x^{100} + 0.5x^{-78} + C \).

Step by step solution

01

Identify the function components

The function given is \( f(x) = 50x^{99} - 39x^{-79} \). It comprises two separate terms: \( 50x^{99} \) and \( -39x^{-79} \). For each term, we will apply the power rule for antiderivatives.
02

Apply the power rule to the first term

The power rule for antiderivatives states that for \( ax^n \), the antiderivative can be found by using \( ax^{n+1}/(n+1) \), provided \( n eq -1 \). Applying this to the first term, \( 50x^{99} \), gives:\[\int 50x^{99} \, dx = \frac{50}{100}x^{100} = 0.5x^{100}\]
03

Apply the power rule to the second term

For the second term, \( -39x^{-79} \), again use the power rule:\[\int -39x^{-79} \, dx = \frac{-39}{-78}x^{-78} = \frac{39}{78}x^{-78} = 0.5x^{-78}\]
04

Combine the antiderivatives

Combine the antiderivatives of the two terms to get the general antiderivative of \( f(x) \):\[ F(x) = 0.5x^{100} + 0.5x^{-78} + C \]where \( C \) is the constant of integration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
The power rule is a fundamental technique for finding antiderivatives. It specifically applies to functions in the form of \( ax^n \), where \( a \) is a constant, and \( n \) is any real number except \(-1\). To find the antiderivative of such a function, we use the formula:\[ \frac{ax^{n+1}}{n+1} \]This rule essentially rolls back the process of differentiation by reversing the power and multiplying by the reciprocal of the new exponent. Here's how it works:
  • Add 1 to the exponent \( n \).
  • Divide the new term by the new exponent \( n+1 \).
The power rule can transform a complex function into a simpler one by changing exponents and adjusting coefficients accordingly. Remember, the power rule cannot be used for \( n = -1 \) because it leads to division by zero.
Integration
Integration is the process of finding the antiderivative of a function. It is the inverse operation to differentiation. When you integrate a function, you are essentially finding the area under its curve on a graph. To integrate a function using antiderivatives, follow these steps:
  • Identify the terms of the function.
  • Apply the appropriate rules, such as the power rule, to each term individually.
  • Combine the antiderivatives of the terms to form the complete antiderivative function.
The function you derive from integration tells you what function, when differentiated, would give you the original function. This makes integration a key tool in calculus for solving problems related to area, velocity, and other real-world applications.
Constants of Integration
When you integrate a function, you often end up adding a constant known as the constant of integration, denoted by \( C \). This constant accounts for any vertical shifts of the function on a graph, as differentiation obliterates any constant information.Why is \( C \) important?
  • It represents an infinite set of antiderivatives, emphasizing that integration's results are not unique.
  • It is crucial for initial value problems where additional conditions specify a unique solution.
In the general solution of an integrated function, \( C \) is a placeholder for all possible vertical translations that could fit the original rate of change described by the differentiated function. Hence, it is standard practice to include \( C \) in every indefinite integral to ensure completeness.

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