Hey guys! Let's dive into the fascinating world of calculus and tackle the integral of 5eˣ * 13x³ * 4x dx. Don't worry, it might seem a bit daunting at first, but we'll break it down into manageable chunks. This guide will walk you through the process step-by-step, making sure you understand every concept along the way. We'll be using some fundamental integration techniques and properties to solve this, so get ready to flex those math muscles! Our main goal is to find the antiderivative of this function, which, when differentiated, will give us back the original function. Ready to get started? Let's go!

Simplifying the Expression: Getting Ready to Integrate

Alright, before we jump into the integration process, let's simplify our expression. We have 5eˣ * 13x³ * 4x dx. First, let's multiply the constants together. We've got 5, 13, and 4. Multiplying these, 5 * 13 * 4 equals 260. We can then rewrite our integral as ∫ 260eˣ * x³ * x dx. Next, we can simplify the variables part using the product rule of exponents (xᵃ * xᵇ = xᵃ⁺ᵇ). We have x³ * x, which can be simplified to x⁴. Now our integral looks like this: ∫ 260eˣx⁴ dx. This is a much cleaner form, and it's easier to work with. Remember, simplifying the expression before integrating is a super important step; it often makes the problem much more manageable. Think of it like tidying up your workspace before starting a project – it makes everything smoother and less confusing! Also, it is very important to remember the constants when multiplying them with variables or simplifying them together because they will not change the integral value. Now that we have simplified our expression, we are ready to find the final result.

Why Simplification Matters

Simplifying expressions before integration is not just about making the problem look nicer; it's a strategic move. It reduces the chances of making errors and can often reveal the most efficient path to the solution. Complex expressions can lead to unnecessary complications and increase the likelihood of overlooking simpler integration techniques. For example, by simplifying constants, you ensure that you are working with the most basic form of the integral. Also, it can help identify patterns or common integration rules that might not be immediately apparent in the original, more complex form. This simplification process is applicable not just in this problem but in all of the problems. It’s like clearing a path through a dense forest to make sure you see the easiest way through. This initial step of simplifying is an integral part of problem-solving in calculus, contributing to both accuracy and efficiency. So, always make sure to simplify before integrating.

Applying Integration by Parts: A Key Strategy

Now, let's get into the main part: integration. We're going to use integration by parts to tackle ∫ 260eˣx⁴ dx. This method is like a clever trick that helps us integrate products of functions. The formula for integration by parts is ∫ u dv = uv - ∫ v du. The goal is to choose our 'u' and 'dv' wisely. Generally, we choose 'u' as the part of the integrand that becomes simpler when differentiated, and 'dv' as the part that is easy to integrate. In our case, let's set u = x⁴ and dv = 260eˣ dx. Now we need to find du and v. Differentiating u = x⁴ gives us du = 4x³ dx. Integrating dv = 260eˣ dx gives us v = 260eˣ. Cool, right? So, now let's plug these values into the integration by parts formula. We'll have ∫ 260eˣx⁴ dx = x⁴ * 260eˣ - ∫ 260eˣ * 4x³ dx.

The Importance of Choosing 'u' and 'dv'

Choosing 'u' and 'dv' correctly is the most important part of applying integration by parts. A good choice can simplify the integral, while a poor choice can lead you down a rabbit hole of increasing complexity. A common mnemonic to help with the selection of 'u' is LIATE or LIPTE, which stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential. The function appearing earlier in this list is typically chosen as 'u'. This strategy guides you to choose 'u' that simplifies when differentiated, making the integral easier. But why does this work? The magic lies in the fact that when we apply integration by parts, we reduce the integral's complexity or make it more familiar. For example, if 'u' is a polynomial (like x⁴), differentiating it will reduce its power or eliminate it entirely. In contrast, 'dv' is chosen to be something we can easily integrate. The goal is always to move toward a simpler integral. Remember, integration by parts is not always the first method that comes to mind, but once you understand how it works, you will be able to solve many complicated integral problems. The method needs to be understood with a lot of practice because there is no one rule to pick 'u' and 'dv'.

Iterating Integration by Parts: Repeated Application

We're not done yet, guys! Notice that we still have ∫ 260eˣ * 4x³ dx. This is another integral that needs to be solved. We'll need to apply integration by parts again! Before we do that, we can simplify this expression to ∫ 1040eˣx³ dx. Now, let's do integration by parts again. Let's set u = x³ and dv = 1040eˣ dx. Differentiating u = x³ gives us du = 3x² dx. Integrating dv = 1040eˣ dx gives us v = 1040eˣ. Plugging these values into our integration by parts formula: ∫ 1040eˣx³ dx = x³ * 1040eˣ - ∫ 1040eˣ * 3x² dx. See the pattern here? We're reducing the power of x with each iteration. We keep applying integration by parts until we get rid of x. The repeated use of integration by parts is a critical skill for solving complex integrals.

When to Stop Iterating

Knowing when to stop iterating integration by parts is crucial to avoid getting stuck in an infinite loop. The key is to keep differentiating the polynomial term ('u') until it simplifies to a constant. This happens when the power of 'x' reaches zero. For example, in our problem, we started with x⁴, then x³, then we will have x². Eventually, we'll get down to just a constant term multiplied by eˣ. Another sign to look for is if the integral becomes something you can easily solve directly, such as a basic exponential integral. Once you reach this point, you can stop applying integration by parts and solve the remaining integral directly. It's also important to note that you should carefully watch for patterns within the integrals. This will help you know when you can stop the process. This understanding is a combination of practice and pattern recognition. Always keep your eye on your goal: simplifying the integral to a form that you can easily solve. This repeated process is complex, and can be easily forgotten, so it is important to understand the concept.

The Final Steps: Bringing it All Together

So, we've applied integration by parts a couple of times. Now, we're going to continue this process until we get rid of all the x terms. After the second application of integration by parts, we have to solve ∫ 1040eˣx³ dx = x³ * 1040eˣ - ∫ 1040eˣ * 3x² dx. And then apply integration by parts again for ∫ 1040eˣ * 3x² dx. Let u = x² and dv = 3120eˣ dx, which gives us du = 2x dx and v = 3120eˣ. So, ∫ 3120eˣx² dx = x² * 3120eˣ - ∫ 6240eˣx dx. Now the next application is ∫ 6240eˣx dx. Let u = x and dv = 6240eˣ dx. So, du = dx and v = 6240eˣ. ∫ 6240eˣx dx = x * 6240eˣ - ∫ 6240eˣ dx. The last integration gives us 6240eˣ. After all the calculations, we can combine all the results. It's time to put all the pieces together and find our final answer. Remember, we started with ∫ 260eˣx⁴ dx. After each application of integration by parts and all the intermediate steps, we get the final result.

The Importance of Correctness and Accuracy

During the final steps of solving integrals, accuracy is really important. There are a few things that can help ensure you're on the right track and to avoid making mistakes. First, double-check your calculations at each step. Make sure you're differentiating and integrating correctly, and that you're using the integration by parts formula accurately. Errors can easily occur, especially when you are repeating a complex process. Second, keep your work organized and label each step clearly. This will help you track your progress and quickly identify any errors. If you have the space to write on paper, make sure you write down everything, because it will help you a lot with the final answer. Third, remember the constant of integration, often denoted as '+ C'. Since you are finding an indefinite integral, always add this constant to the end of your final answer. This constant represents all the possible antiderivatives of the function. Fourth, after you've solved the integral, you can verify your answer by differentiating it. Differentiating your answer should give you the original function (260eˣx⁴), which confirms that your solution is correct. If the result is the same function, then it is correct. All of these tips will guide you in problem-solving and make your work more accurate and correct.

The Final Answer

Okay, guys! After all that hard work and going through all those steps, the integral of 5eˣ * 13x³ * 4x dx is: 260eˣx⁴ - 1040eˣx³ + 3120eˣx² - 6240ex + 6240eˣ + C. We have successfully found the antiderivative of our original function! Yay! We have applied integration by parts multiple times, simplified the expressions, and made sure that we didn't forget the constant of integration. It was a long journey, but hopefully, you've learned a lot and gained a better understanding of how to solve these kinds of integrals. Remember, practice is key, so keep working through different types of problems and you will become more comfortable with these techniques.

Tips for Future Practice

To become more proficient in solving integrals like the one we just worked through, here are a few things to keep in mind: First, practice regularly. The more problems you solve, the more familiar you will become with different techniques and patterns. Secondly, start with simpler problems to build your confidence and understanding of the fundamental concepts. Next, make sure you understand the integration by parts formula and when to use it, because it is one of the most important methods. Then, always try to simplify the expression before you start integrating. This can make the process much easier. Moreover, check your answers by differentiating them. This will help you identify any errors in your work. Also, learn to recognize patterns in integrals. This will allow you to choose the correct strategy more quickly. Finally, don't be afraid to ask for help from classmates or your teacher. It's always helpful to get a different perspective or explanation. With consistent effort and the right approach, you'll be able to conquer any integral that comes your way! Keep practicing and you'll become a calculus pro in no time! Good luck! And congratulations on solving the integral!