Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int x^{-4} d x$$

Step-by-Step Solution

Answer

The indefinite integral of \(x^{-4}\) is \(\int{x^{-4}dx} = \frac{-1}{3x^3} + C\).

Step 1: Identify the function to integrate

We are given the function \(x^{-4}\) to integrate.

Step 2: Apply the power rule for integration

The power rule for integration states that: \[\int{x^n dx} = \frac{x^{n+1}}{n+1} + C\] In our case, \(n = -4\).

Step 3: Evaluate the integral

Applying the power rule to our function, we get: \[\int{x^{-4} dx} = \frac{x^{-4+1}}{-4 + 1} + C\] Now simplifying the expression, we have: \[\int{x^{-4} dx} = \frac{x^{-3}}{-3} + C\] Thus, the indefinite integral of the given function is: \[\int{x^{-4} dx} = \frac{-1}{3x^3} + C\]

Key Concepts

Power Rule for Integration

The power rule is one of the most fundamental techniques in the arsenal of integral calculus. What does it entail? Simply put, if you need to integrate a power function of the form \(x^n\), where \(n\) is any real number except -1, the power rule can be your guiding star. It provides us with a straightforward formula:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]Here, \(C\) is the constant of integration, a critical component since an indefinite integral represents a family of functions.

• The rule is easy to apply, but care must be taken not to use it when \(n = -1\) since that would lead to division by zero, which isn’t defined.
• It is crucial to remember to add one to the exponent and then divide by this new exponent.
• This rule emerges from the reverse process of differentiation where the derivative of \(x^{n+1}\) with respect to \(x\) is \(n+1\) multiplied by \(x^n\).

By understanding and applying the power rule correctly, you are equipping yourself to solve a vast array of integral problems with confidence and precision.

Integral Calculation

Integral calculation is a cornerstone of calculus, involving the process of finding the integral of a function. But why does integral calculation matter? It serves as a means to,

• Determine the accumulated quantity, such as area under a curve, from a rate of change.
• Reverses the process of differentiation, making it possible to recover an original function from its derivative.
• Address problems in physics, engineering, economics, and beyond, where continuous change is involved.

Approaching the Integral Calculation

• Identify the type of function you are dealing with – is it polynomial, trigonometric, exponential?
• Detect if there are known integration formulas or rules, like the power rule, that can be applied.
• Simplify the function if possible – can it be broken into simpler parts?
• Perform the integration step by step, always including the constant of integration \(C\) at the end for indefinite integrals.
• Verify your work by differentiating the result and checking if you obtain the original function.

The ability to calculate integrals accurately opens the door to understanding a vast range of phenomena across various sciences. It’s a tool for translating the language of the universe into a form we can comprehend and work with.

Integral of a Power Function

Integrating a power function may seem daunting, but it's just another one of those tasks that requires methodical execution. The power function, expressed as \(f(x) = x^n\), where \(n\) is a real number, has a definite pattern when integrated. Here is how we approach the integral of a power function:

• First, ensure that you're not dealing with the special case where \(n = -1\).
• Next, apply the power rule for integration, as this type of function is tailor-made for such a rule.
• Don’t forget to include the constant of integration, \(C\), which is essential as multiple functions could have the same derivative and thus integral.

When dealing with negative exponents, such as in the example \(\int x^{-4} dx\), some might feel a moment of hesitation. Fear not—the same rule applies:\[\int x^{-4} \, dx = \frac{x^{-3}}{-3} + C\]Notably, we've used a negative exponent here. The calculation confirms that integrating power functions, whether they have positive or negative exponents, involves the same procedure. Understanding the mechanics of such integrals is immensely helpful, as power functions frequently arise in various mathematical contexts. Knowledge of their integration process is not just an academic exercise but a practical tool in solving real-world problems.