Evaluating ∫₁⁴ (x³/² + 4) / X² Dx: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of calculus to tackle the integral ∫₁⁴ (x³/² + 4) / x² dx. Don't worry if this looks a bit intimidating at first. We'll break it down step by step, making sure it's super clear and easy to follow. So, grab your pens and let's get started on this exciting mathematical journey! This guide is designed to help you, whether you're a seasoned mathematician or just starting out with calculus. We'll cover all the necessary techniques to solve this integral. Let's transform this into a simpler form so we can easily perform the integration. Our main goal is to evaluate this definite integral. Let's get straight to it and see how this integral unwinds itself.

Simplifying the Integral Expression: A First Step

Alright guys, the first thing we're going to do is simplify the integral's expression. Remember, the integral is ∫₁⁴ (x³/² + 4) / x² dx. To make things easier, let's split the fraction into two separate terms. This way, we can deal with each part individually. This is a super common trick in calculus, and it's going to make our lives a lot easier. It's like taking apart a complex puzzle to solve it piece by piece. Here's what it looks like when we split it up: ∫₁⁴ (x³/² / x² + 4 / x²) dx. See? Much cleaner already! Now, let's focus on simplifying each of these terms further before we do anything else. We want to make sure everything is in its simplest form so we can apply our integration rules without a hitch. Don't skip this step; it makes a huge difference in how straightforward the rest of the process will be. Remember, guys, the goal here is to make the problem less intimidating and more manageable. The key is to remember your exponent rules. Keep in mind that when dividing exponents with the same base, you subtract the powers. Okay, let's simplify that first term, x³/² / x². Using the rule for dividing exponents, we subtract the powers: 3/2 - 2 = -1/2. So, x³/² / x² simplifies to x⁻¹/². For the second term, 4 / x², we can rewrite this as 4x⁻². This is just another way to represent the same thing. So, now our integral looks like this: ∫₁⁴ (x⁻¹/² + 4x⁻²) dx. See? It's already looking less scary! This step is all about making the integral user-friendly before we start integrating. It's like prepping your ingredients before cooking a meal—everything runs smoother and more efficiently. With these simplifications, we're setting ourselves up for a successful integration, which is what this whole exercise is about. We'll be using the power rule for integration next. The power rule is our go-to tool for handling terms like x raised to a power.

Applying the Power Rule for Integration

Now that we've simplified our integral, it's time to apply the power rule for integration. This rule is a fundamental concept in calculus. It helps us find the antiderivative of functions that are in the form of x raised to a power, like the ones we have here. The power rule states that the integral of xⁿ dx is (xⁿ⁺¹ / (n+1)) + C, where C is the constant of integration. Let's apply this rule to our simplified integral: ∫₁⁴ (x⁻¹/² + 4x⁻²) dx. First, let's integrate x⁻¹/². Using the power rule, we add 1 to the exponent: -1/2 + 1 = 1/2. Then, we divide by the new exponent: x¹/² / (1/2). Simplify, and you get 2x¹/². Next, let's integrate 4x⁻². Apply the power rule again: -2 + 1 = -1. Then divide by the new exponent: 4x⁻¹ / -1, which simplifies to -4x⁻¹. So, the integral of 4x⁻² is -4x⁻¹. Therefore, the antiderivative of our integral ∫₁⁴ (x⁻¹/² + 4x⁻²) dx is 2x¹/² - 4x⁻¹. Remember, the antiderivative is the function whose derivative is equal to the original function. To complete the definite integral, we will substitute the limits of integration into the antiderivative and subtract the results. This is an essential step in evaluating definite integrals, as it gives us a numerical value representing the area under the curve of the function between the given limits. The antiderivative we obtained, 2x¹/² - 4x⁻¹, is a critical result since we can use it to evaluate our definite integral between the limits of 1 and 4. The power rule is not only fundamental but also an easy method, so it's important that you master it to fully understand this integral. This method is one of the most used techniques. So, keep this in mind and practice some more to fully understand it.

Evaluating the Definite Integral: The Final Step

We're almost there, guys! Now that we've found the antiderivative, we need to evaluate the definite integral. This means plugging in the upper and lower limits of integration into the antiderivative and subtracting the results. Our antiderivative is 2x¹/² - 4x⁻¹, and our limits of integration are 1 and 4. First, let's plug in the upper limit, 4: 2(4)¹/² - 4(4)⁻¹. This simplifies to 2(2) - 4(1/4), which equals 4 - 1 = 3. Next, let's plug in the lower limit, 1: 2(1)¹/² - 4(1)⁻¹. This simplifies to 2(1) - 4(1), which equals 2 - 4 = -2. Finally, we subtract the result from the lower limit from the result from the upper limit: 3 - (-2) = 3 + 2 = 5. So, the value of the definite integral ∫₁⁴ (x³/² + 4) / x² dx is 5. This means that the area under the curve of the function from x = 1 to x = 4 is 5 units. This is our final result! Evaluating definite integrals is a critical skill in calculus. It allows us to find the precise area under a curve, which is super useful in a variety of applications, from physics to economics. The area under the curve can represent various quantities such as the displacement of an object, the total work done, or the total amount of a substance. The process involves finding the antiderivative of the function and then calculating the difference between the antiderivative's values at the upper and lower limits of integration. This provides a complete picture of the function's behavior within the specified interval. Be sure to practice more examples so you can master all the steps.

Conclusion

And that's a wrap, folks! We successfully evaluated the integral ∫₁⁴ (x³/² + 4) / x² dx, and our answer is 5. We started by simplifying the integral, then applied the power rule for integration, and finally evaluated the definite integral using the limits of integration. Each step was designed to be as clear as possible, making sure you understood not just what we did, but also why we did it. This example perfectly illustrates the fundamental principles of integral calculus and how to apply them to solve complex problems. Remember, the more you practice, the better you'll get. Keep at it, and you'll find that calculus, like any other skill, becomes more manageable and even enjoyable over time. I encourage you to try solving more integrals on your own. Maybe try similar problems but change the limits of integration or the function itself. Playing around with these parameters will help you understand the concepts better. Thanks for joining me on this calculus adventure! I hope this helps you succeed in your math journey and helps you discover the beauty of calculus. Remember that calculus is all about understanding the relationships between things. And if you have any questions, don't hesitate to ask. Happy calculating!