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Chapter 6: Problem 10

Evaluate the definite integrals. \(\int_{-2}^{-1}\left(x^{-5}+1\right) d x\)

Short Answer

Expert verified
\( \frac{49}{64} \)

Step by step solution

01

Identify the Integral Components

The integral to evaluate is \( \int_{-2}^{-1} (x^{-5} + 1) \, dx \). This is the sum of two separate integrals: \( \int_{-2}^{-1} x^{-5} \, dx \) and \( \int_{-2}^{-1} 1 \, dx \).
02

Apply the Power Rule to \( x^{-5} \)

For \( \int x^{-5} \, dx \), use the power rule, which states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \). Here, \( n = -5 \), so \( \int x^{-5} \, dx = \frac{x^{-4}}{-4} = -\frac{1}{4} x^{-4} \).
03

Integrate the Constant Function

For \( \int 1 \, dx \), the antiderivative is \( x \). Therefore, \( \int_{-2}^{-1} 1 \, dx = x \Big|_{-2}^{-1} = -1 - (-2) = 1 \).
04

Evaluate \( \int_{-2}^{-1} x^{-5} \, dx \)

Using the antiderivative from Step 2, evaluate \( -\frac{1}{4} x^{-4} \) from \(-2\) to \(-1\). Substituting these limits gives \(-\frac{1}{4}(-1)^{-4} - (-\frac{1}{4}(-2)^{-4}).\)
05

Simplify the Evaluated Function

Calculate \(-\frac{1}{4}(-1)^{-4} = -\frac{1}{4}(-1) \). Simplifying, this equals \(-\frac{1}{4} \).ewlineFor \(-\frac{1}{4}(-2)^{-4} = -\frac{1}{4}\times\frac{1}{16} = -\frac{1}{64} \). Thus, the result is \(-\frac{1}{4} - (-\frac{1}{64}) = -\frac{1}{4} + \frac{1}{64} = -\frac{16}{64} + \frac{1}{64} = -\frac{15}{64}.\)
06

Combine the Results

The total evaluated integral is \(-\frac{15}{64} + 1\). Converting 1 to a denominator of 64, we have \(\frac{64}{64}\). Thus, the final result is \( \frac{49}{64} \).

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Key Concepts

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

Power Rule
When dealing with integrals, the power rule is a fundamental tool that simplifies the process of finding antiderivatives. The power rule for integration is expressed as:
  • \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)
  • "n" is any real number except -1.
In the power rule, "C" represents the constant of integration, but for definite integrals, it is not necessary. You directly compute the definite integral by substituting the limits. This method is particularly useful because it provides a straightforward way to handle polynomials and similar expressions. It allows us to quickly find antiderivatives without needing to guess or use advanced techniques.
Integration
Integration is the process of finding the integral of a function, and it can be thought of as the reverse operation to differentiation. When we integrate a function, we are essentially looking for the original function that, when differentiated, will yield the primary function.
  • Definite Integrals are calculated between two limits, providing us with a specific value.
  • This result can be interpreted as the net area under the curve, from the lower to the upper limit.
In definite integrals, the integration process not only helps in finding areas but also aids in solving problems related to total accumulation, such as distance, mass, or volume. It is a powerful tool across various fields like physics, engineering, and statistics.
Antiderivative
An antiderivative of a function is a function whose derivative returns the original function. When evaluating definite integrals, finding the antiderivative is a crucial step. For instance, in our exercise:
  • The antiderivative of \( x^{-5} \) is \( -\frac{1}{4}x^{-4} \).
  • For constant functions like 1, the antiderivative is simply \( x \).
The process of integration often involves finding these antiderivatives, which then allow us to compute the value of definite integrals by evaluating the antiderivative at the given limits. By determining antiderivatives accurately, we can solve many real-world problems involving rates of change and cumulative processes.
Step-by-step Solution
The step-by-step approach to solving definite integrals ensures we understand each part of the integration process. In our example, we tackled the integral \( \int_{-2}^{-1}(x^{-5} + 1) \, dx \) by breaking it down:
  • First, separating it into two integrals: \( \int_{-2}^{-1}x^{-5} \, dx \) and \( \int_{-2}^{-1} \, dx \).
  • Then, applying the power rule to find the antiderivative for \( x^{-5} \).
  • Using straightforward algebra for the constant function 1.
  • Recovering the antiderivatives at both limits and calculating the difference.
  • Finally, combining these values to achieve the definite result.
This structured methodology helps clarifies the problem-solving process, ensuring no step is overlooked and making it easier for learners to follow and comprehend the underlying concepts.

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