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Chapter 17: Problem 4

Find \(\int 7 x^{-2} \mathrm{~d} x\).

Short Answer

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Question: Find the integral of the function \(7x^{-2}\) with respect to x. Answer: The integral of \(7x^{-2}\) with respect to x is \( -7x^{-1} + C\).

Step by step solution

01

Identify the function and its exponent

The given function is \(7x^{-2}\). Here, the exponent is -2.
02

Apply the power rule for integration

Using the power rule, we have: 鈭玕(7x^{-2}\) dx = \(7\int x^{-2} dx\) Now, by the power rule, we have: \(7\int x^{-2} dx = 7 \cdot \frac{x^{-2 + 1}}{-2 + 1} + C\)
03

Simplify the expression

Now, we will simplify the expression as follows: \(= 7 \cdot \frac{x^{-1}}{-1} + C\) \(= -7x^{-1} + C\)
04

Write the final answer

The integral of \(7x^{-2}\) with respect to x is: \( 鈭� 7x^{-2} dx = -7x^{-1} + C\)

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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 your go-to tool for functions expressed with exponents. It helps you easily integrate functions of the form \(x^n\). The rule is straightforward: increase the exponent by one, then divide by the new exponent. This transforms \(\int x^n \, dx\) into \(\frac{x^{n+1}}{n+1} + C\).

Let's break down the example of \(7x^{-2}\). Here, you view the function as \(x^{-2}\), apply the rule, and change the exponent to \(-1\). Then, divide by \(-1\) and don't forget to multiply by the constant 7 at the front. These consistent steps yield the integral.
Exponents in Calculus
Exponents play a crucial role in calculus, especially when performing integration. Here鈥檚 why understanding them is key:
  • They dictate the curve鈥檚 steepness or flatness.
  • Negative exponents can transform the function, flipping its direction.
  • In integration, adjusting exponents by one ensures a smooth transition from differentiation back to original form.
For \(x^{-2}\), the negative exponent signifies a fractional decrease and an inverse relationship in the function鈥檚 growth. This awareness is vital as you manipulate functions during integration.
Definite vs Indefinite Integrals
Integrals are essential in finding areas under curves or determining accumulations. They are split into two types:
  • Indefinite Integrals: Have the general form \(\int f(x) \, dx\), with a result that includes \(C\), the constant of integration. They represent a family of functions.
  • Definite Integrals: Involve limits \([a, b]\) and yield a numerical value, representing the area under the curve from \(a\) to \(b\).
In our example, \(\int 7x^{-2} \, dx\) is an indefinite integral. It doesn鈥檛 have bounds, and so its result is a function rather than a specific number, demonstrated by \(-7x^{-1} + C\).
Constant of Integration
Whenever you encounter an indefinite integral, you鈥檒l need to include a constant of integration, denoted by \(C\). Here鈥檚 why it鈥檚 important:
  • Represents an infinite possibility of vertical shifts of the function鈥檚 graph.
  • Accounts for every possible antiderivative of the function.
  • Ensures that no solution is overlooked in broader contexts, like solving differential equations.
The constant \(C\) enriches our solution \(-7x^{-1} + C\), reminding us that while the specific form is determined, the exact position might vary depending on additional conditions or information.

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