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Chapter 4: Problem 5

Give the antiderivatives of \(x^{p}\). For what values of \(p\) does your answer apply?

Short Answer

Expert verified
Answer: The antiderivative of \(x^p\), where \(p\) is any constant, is given by the formula: \(\int x^p dx = \frac{x^{p+1}}{p+1} + C\) for \(p \neq -1\) For \(p = -1\), the antiderivative is: \(\int x^{-1} dx = \ln |x| + C\)

Step by step solution

01

Understand the function

Given, \(x^{p}\), where \(p\) is any constant. We need to find its antiderivative.
02

Apply the power rule for antiderivatives

The power rule for antiderivatives states that the antiderivative of \(x^n\) is given by \(\frac{x^{n+1}}{n+1} + C\), where \(n\) is any constant, and \(C\) is the constant of integration. So, in our case, the antiderivative of \(x^{p}\) is: \(\int x^p dx = \frac{x^{p+1}}{p+1} + C\)
03

Determine the values of p for which the formula applies

This formula for antiderivatives applies for any value of \(p\) except for \(p = -1\). If \(p = -1\), then the denominator will become zero, which is not allowed since division by zero is undefined. In the case of \(p = -1\), the integral of \(x^{-1}\) is the natural logarithm function, given by: \(\int x^{-1} dx = \ln |x| + C\)
04

Summarize the results

The antiderivative of \(x^p\) is given by the formula: \(\int x^p dx = \frac{x^{p+1}}{p+1} + C\) for \(p \neq -1\) For \(p = -1\), the antiderivative is: \(\int x^{-1} dx = \ln |x| + C\)

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