Solving ∫ (x³ - 1)² / X² Dx A Step-by-Step Guide

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Hey guys! Today, we're diving into a fun calculus problem: solving the integral of (x³ - 1)² / x² dx. This might look a bit intimidating at first, but don't worry, we'll break it down step-by-step and make it super easy to understand. So, grab your pencils and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the question is asking. We need to find the integral of the function (x³ - 1)² / x² with respect to x. In simpler terms, we're looking for a function whose derivative is (x³ - 1)² / x². Integrals are a fundamental concept in calculus, and mastering them opens doors to solving a wide range of problems in physics, engineering, and other fields. This particular integral involves a polynomial function divided by another polynomial function, which means we'll need to use some algebraic manipulation and the power rule for integration to solve it effectively. Understanding the problem thoroughly is the first step towards finding the correct solution, and it also helps in appreciating the underlying concepts of calculus. So, let's move on to the next step: expanding the numerator.

Step 1: Expanding the Numerator

The first thing we need to do is simplify the expression inside the integral. The numerator, (x³ - 1)², looks like it could be expanded. Let's do that using the formula (a - b)² = a² - 2ab + b². In our case, a = x³ and b = 1. So, expanding (x³ - 1)² gives us:

(x³ - 1)² = (x³)² - 2(x³)(1) + 1² = x⁶ - 2x³ + 1

Now our integral looks like this:

∫ (x⁶ - 2x³ + 1) / x² dx

Expanding the numerator is a crucial step because it allows us to separate the complex fraction into simpler terms that are easier to integrate. This algebraic manipulation is a common technique in calculus, and it's essential to be comfortable with such operations. By expanding the numerator, we've transformed the original problem into a more manageable form, paving the way for the next step: dividing each term by the denominator. This methodical approach is key to tackling more complex integrals and ensuring accuracy in our calculations. So, let's proceed to the next step and see how dividing by the denominator simplifies the problem further.

Step 2: Dividing Each Term by the Denominator

Now that we've expanded the numerator, we can divide each term by the denominator, x². This will break the integral into simpler fractions that we can easily integrate using the power rule. So, we have:

(x⁶ - 2x³ + 1) / x² = x⁶ / x² - 2x³ / x² + 1 / x²

Using the rules of exponents (xᵃ / xᵇ = xᵃ⁻ᵇ), we can simplify each term:

x⁶ / x² = x⁴ -2x³ / x² = -2x 1 / x² = x⁻²

So our integral now becomes:

∫ (x⁴ - 2x + x⁻²) dx

Dividing each term by the denominator is a significant step in simplifying the integral. This algebraic manipulation allows us to transform a complex fraction into a sum of simpler terms, each of which can be integrated individually using the power rule. By breaking down the problem into smaller, more manageable parts, we make the integration process much easier and less prone to errors. This technique is widely used in calculus and is essential for solving a variety of integral problems. Now that we have a simplified expression, we can move on to the next step: integrating each term separately. Let's see how the power rule helps us find the antiderivative of each term.

Step 3: Integrating Each Term Separately

Now we have a much simpler integral: ∫ (x⁴ - 2x + x⁻²) dx. We can integrate each term separately using the power rule for integration, which states that ∫xⁿ dx = (xⁿ⁺¹) / (n + 1) + C, where C is the constant of integration.

Let's apply the power rule to each term:

∫ x⁴ dx = (x⁴⁺¹) / (4 + 1) + C₁ = x⁵ / 5 + C₁ ∫ -2x dx = -2 ∫ x¹ dx = -2(x¹⁺¹) / (1 + 1) + C₂ = -2(x²) / 2 + C₂ = -x² + C₂ ∫ x⁻² dx = (x⁻²⁺¹) / (-2 + 1) + C₃ = x⁻¹ / (-1) + C₃ = -1/x + C₃

So, the integral of each term is:

x⁵ / 5 for x⁴ -x² for -2x -1/x for x⁻²

Integrating each term separately is a key technique in calculus, especially when dealing with sums or differences of functions. The power rule is a fundamental tool for integrating polynomial terms, and understanding how to apply it correctly is crucial. By breaking down the integral into individual terms, we can focus on each part and find its antiderivative more easily. Remember to include the constant of integration, C, for each term, as this accounts for the family of functions that have the same derivative. Now that we have integrated each term, the final step is to combine the results and simplify the expression. Let's move on to the next section and see how we can write the final answer.

Step 4: Combining the Results and Adding the Constant of Integration

Now that we've integrated each term, we need to combine the results and add the constant of integration, C. This constant represents the family of functions that have the same derivative as the integrand. So, combining our results from the previous step, we get:

∫ (x⁴ - 2x + x⁻²) dx = x⁵ / 5 - x² - 1/x + C

Here, C represents the sum of the constants of integration from each term (C₁ + C₂ + C₃). It's important to include the constant of integration because the derivative of a constant is always zero, meaning there could be an infinite number of constant terms that would result in the same derivative. Therefore, we represent this uncertainty with the constant C.

Combining the results and adding the constant of integration is the final step in solving the integral. This step ensures that we have accounted for all possible antiderivatives and provides the most general solution to the problem. By including the constant C, we acknowledge the fact that the antiderivative is not unique and represents a family of functions. This understanding is crucial for applying integrals in real-world problems, where initial conditions or boundary conditions are often needed to determine the specific value of the constant C. With the final answer in hand, let's take a moment to review the steps we took and appreciate the process of solving this integral.

Final Answer

So, the final answer to the integral ∫ (x³ - 1)² / x² dx is:

x⁵ / 5 - x² - 1/x + C

Where C is the constant of integration. Yay! We did it!

Conclusion

We've successfully solved the integral! Remember, the key steps were:

  1. Expanding the numerator to simplify the expression.
  2. Dividing each term by the denominator to break the integral into smaller parts.
  3. Integrating each term separately using the power rule.
  4. Combining the results and adding the constant of integration.

This problem demonstrates how algebraic manipulation and the power rule can be used to solve integrals involving polynomial functions. Remember to practice these techniques, guys, and you'll become integral masters in no time! Calculus might seem challenging at first, but with a methodical approach and consistent practice, you can conquer any integral. Keep exploring, keep learning, and most importantly, keep having fun with math!

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