Step-by-Step Guide To Solve Integral Of (∛x³ - ⁴√x³) Dx

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Hey guys! Today, we're diving into a fun and interesting calculus problem: solving the integral of (∛x³ - ⁴√x³) dx. This integral might look a bit intimidating at first glance, but don't worry, we'll break it down step by step. We'll use some basic algebraic manipulations and integral rules to make it super easy to understand. So, grab your pencils and notebooks, and let's get started!

Understanding the Integral

Before we jump right into solving this integral, let’s make sure we really understand what we’re dealing with. When we talk about integrals, we’re essentially discussing the reverse process of differentiation. Think of it as finding the area under a curve. This particular integral, ∫(∛x³ - ⁴√x³) dx, involves two terms inside the integral sign: ∛x³ and ⁴√x³. Our mission is to find the antiderivative of this expression. So, why is this important? Well, integrals have tons of applications in various fields, from physics (calculating displacement and work) to engineering (designing structures) and even economics (modeling growth). Getting comfortable with integrals like this one is a fantastic way to build a solid foundation in calculus. The key to tackling any integral is to first simplify the expression inside the integral sign. This often involves rewriting radicals as fractional exponents, which will make the integration process much smoother. We’ll be using the power rule for integration, which states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n is any real number except -1, and C is the constant of integration. So, keep this rule in mind as we go through the steps. We will also discuss how to deal with the constant of integration, which is a crucial part of indefinite integrals. We need to remember that when we find an antiderivative, there are infinitely many possible solutions that differ by a constant. The constant of integration, C, represents this family of functions. So, by the end of this guide, you'll not only know how to solve this specific integral but also understand the underlying principles and techniques that you can apply to a wide range of calculus problems. Ready to dive in? Let’s go!

Step 1: Simplify the Expression

Okay, the first thing we need to do is simplify the expression inside the integral. We’ve got ∛x³ and ⁴√x³, and these radical forms can be a bit tricky to work with directly. The trick here is to rewrite these radicals using fractional exponents. Remember that ∛x is the same as x^(1/3) and ⁴√x is the same as x^(1/4). So, ∛x³ can be rewritten as (x³)^(1/3), and ⁴√x³ becomes (x³)^(1/4). This simple change makes a huge difference because it allows us to use the properties of exponents to further simplify things. Now, let's apply the power of a power rule, which says that (am)n = a^(m*n). For our first term, (x³)^(1/3), we multiply the exponents: 3 * (1/3) = 1. So, ∛x³ simplifies to x¹. Or just x, nice and clean! For the second term, (x³)^(1/4), we again multiply the exponents: 3 * (1/4) = 3/4. So, ⁴√x³ simplifies to x^(3/4). Now our integral looks much more manageable. We’ve transformed ∫(∛x³ - ⁴√x³) dx into ∫(x - x^(3/4)) dx. See how much simpler that is? This step is crucial because it turns a complex-looking problem into something we can easily handle using basic integration rules. We've taken the original expression, which was a bit messy with its cube roots and fourth roots, and streamlined it into a form that's ready for integration. This simplification technique is a cornerstone of solving many calculus problems, so mastering it is super important. By rewriting the radicals as fractional exponents, we’ve opened the door to applying the power rule for integration, which we'll use in the next step. So, always remember: when you see radicals in an integral, think about fractional exponents. It's your best friend for simplifying expressions and making your calculus life a whole lot easier. Now that we've got our simplified expression, we’re all set to move on to the next step: actually integrating the terms. Let's do it!

Step 2: Apply the Power Rule for Integration

Alright, now that we've simplified our integral to ∫(x - x^(3/4)) dx, it’s time to actually integrate! This is where the power rule for integration comes into play. As a quick refresher, the power rule states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where n is any real number except -1, and C is the constant of integration. So, we're going to apply this rule to each term in our integral separately. First up, we have ∫x dx. Here, x is the same as x¹, so n = 1. Using the power rule, we add 1 to the exponent and divide by the new exponent: ∫x¹ dx = (x¹⁺¹)/(1+1) = x²/2. Easy peasy! Next, let's tackle ∫x^(3/4) dx. In this case, n = 3/4. Applying the power rule, we get: ∫x^(3/4) dx = (x^(3/4 + 1))/(3/4 + 1) = (x^(7/4))/(7/4). Now, dividing by a fraction is the same as multiplying by its reciprocal, so we can rewrite this as (4/7)x^(7/4). Fantastic! Now we've integrated both terms separately. Our integral ∫(x - x^(3/4)) dx becomes x²/2 - (4/7)x^(7/4). But wait, there’s one more super important thing we need to remember: the constant of integration, C. Since we're dealing with an indefinite integral (meaning we don't have specific limits of integration), we need to add C to our result. This constant represents the fact that the derivative of any constant is zero, so when we reverse the process (integration), we need to account for all possible constant terms. So, our final result after applying the power rule is x²/2 - (4/7)x^(7/4) + C. We've successfully integrated each term and remembered our constant of integration. This is a huge step! We’ve taken the simplified expression and used the power rule to find its antiderivative. This step really highlights the power and versatility of the power rule, which is one of the most fundamental tools in integral calculus. By applying this rule carefully and remembering the constant of integration, we’re well on our way to solving the entire problem. Next up, we’ll just tidy things up a bit and present our final answer in a clear and understandable way. Let’s keep going!

Step 3: Simplify and Present the Final Answer

Okay, we've done the heavy lifting! We’ve integrated the expression and arrived at x²/2 - (4/7)x^(7/4) + C. Now, it's time to simplify our answer and present it in a nice, clear format. Sometimes, just tidying up the expression can make a big difference in how easy it is to understand and use. In this case, our expression is already pretty clean, but let's just make sure everything is in its simplest form. We have x²/2, which is straightforward. And we have (4/7)x^(7/4). The fractional exponent x^(7/4) can also be rewritten in radical form if we want to. Remember that x^(7/4) is the same as ⁴√(x⁷). So, we could rewrite (4/7)x^(7/4) as (4/7)⁴√(x⁷). Whether you prefer the fractional exponent or the radical form is often a matter of personal preference or what the specific problem requires. Both forms are mathematically correct. The most important thing is to present your answer in a way that is clear and easy to read. And, of course, we can’t forget our constant of integration, C. It’s a crucial part of the answer for indefinite integrals. So, our final answer can be presented in two equivalent forms:

  1. x²/2 - (4/7)x^(7/4) + C
  2. x²/2 - (4/7)⁴√(x⁷) + C

Both of these are perfectly acceptable answers. The first form, with the fractional exponent, is often preferred in more advanced calculus contexts because it's easier to manipulate algebraically. The second form, with the radical, might be more intuitive for some people to visualize. So, there you have it! We’ve successfully solved the integral of (∛x³ - ⁴√x³) dx. We started by simplifying the expression using fractional exponents, then applied the power rule for integration, and finally tidied up our answer and included the constant of integration. This step is all about making sure our hard work pays off by presenting a final answer that’s both correct and clear. Think of it as putting a bow on a beautifully wrapped gift! We've shown all the steps, explained the reasoning behind each one, and given you two ways to express the final answer. This should give you a solid understanding of how to tackle similar problems in the future. Remember, practice makes perfect, so try working through a few more integrals on your own to really solidify these concepts. And always double-check your work and make sure your final answer is presented in a way that’s easy to understand. You’ve got this!

Key Takeaways

So, what have we learned today, guys? We've tackled a seemingly complex integral and broken it down into manageable steps. The key takeaways from this exercise are super valuable for your calculus journey. First, we learned the importance of simplification. Rewriting radicals as fractional exponents is a game-changer. It transforms intimidating expressions into forms that are much easier to work with. This is a trick you’ll use again and again in calculus, so make sure you’ve got it down. Next up, we really hammered home the power rule for integration. This is one of the most fundamental tools in your calculus toolbox. Mastering the power rule allows you to integrate a wide range of expressions, and we saw exactly how to apply it step by step. We also emphasized the crucial role of the constant of integration, C. Never forget to add C when you're dealing with indefinite integrals! It represents the family of functions that have the same derivative, and it’s an essential part of a complete and correct answer. Beyond the specific steps, we also highlighted the importance of understanding the process. Calculus isn't just about memorizing formulas; it's about understanding why those formulas work and how to apply them in different situations. By walking through each step slowly and explaining the reasoning, we aimed to give you a deeper understanding of integral calculus. And finally, we stressed the value of presenting your answer clearly. A correct answer that's messy and hard to read isn't as useful as a well-presented solution. Taking the time to tidy up your work and express your answer in a clear format shows that you understand the material and can communicate your results effectively. So, remember these key takeaways: simplify, apply the power rule, include the constant of integration, understand the process, and present your answer clearly. These are the building blocks of success in calculus, and they'll serve you well as you tackle more challenging problems. Keep practicing, keep exploring, and most importantly, keep enjoying the journey of learning calculus! You've got this, and we're here to help you every step of the way.

Practice Problems

Now that we’ve walked through solving the integral of (∛x³ - ⁴√x³) dx step by step, it's time to put your new skills to the test! Practice is absolutely crucial for mastering calculus, so let’s dive into a few problems that are similar to what we’ve just covered. These problems will help you solidify your understanding of simplifying expressions, applying the power rule for integration, and remembering that all-important constant of integration. Try to work through these problems on your own first. This will give you a good sense of where you’re strong and where you might need a little more practice. Don't worry if you get stuck – that’s totally normal! The important thing is to make an effort and learn from the process. After you’ve given each problem a try, you can check your answers and solutions to see how you did. And if you're still feeling unsure about something, go back and review the steps we covered earlier in this guide. Remember, calculus is a journey, and every problem you solve is a step forward. So, let’s get started! Here are a few practice problems to try:

  1. ∫(√x - ∛x) dx
  2. ∫(⁵√x² + x^(3/2)) dx
  3. ∫(x⁴ - ⁶√x⁵) dx

For each of these problems, remember to start by simplifying the expressions inside the integral. Rewrite any radicals as fractional exponents, and then apply the power rule for integration. Don’t forget to add the constant of integration, C, to your final answer! Once you’ve solved these problems, you’ll have a much stronger grasp of the techniques we’ve discussed. And the more you practice, the more confident you’ll become in your calculus abilities. These practice problems are designed to reinforce the concepts we’ve covered and give you the opportunity to apply them in different contexts. Each problem presents a slightly different twist, so working through them will help you develop a deeper understanding of the material. And remember, if you’re struggling with any of these problems, don’t hesitate to seek out help. There are tons of resources available, from online tutorials to textbooks to your classmates and instructors. The key is to keep practicing and keep learning. So, grab your pencils, notebooks, and calculators, and let’s tackle these practice problems. You’ve got the tools, you’ve got the knowledge, and now it’s time to put it all into action. Happy solving!