Hey there, math enthusiasts! Ever stumbled upon an integral that seems a bit… daunting? Let's break down the integral of 8 * sqrt(4x^7) dx. We'll walk through it step by step, making sure everyone can follow along. No need to be intimidated; it's all about understanding the process.

Understanding the Problem: The Basics of Integration

Alright, guys, let's start with the basics. Integration is, essentially, the opposite of differentiation. Think of it as finding the area under a curve. When we see ∫ f(x) dx, we're being asked to find the antiderivative of the function f(x). In our case, f(x) is 8 * sqrt(4x^7). This means we need to find a function whose derivative is 8 * sqrt(4x^7). Sounds fun, right? Don't worry, it's not as scary as it sounds. We'll get through it together.

Now, before we jump into the integral, let's make sure we're on the same page with a few key concepts. Remember the power rule for integration? It's our best friend here. The power rule states that the integral of x^n is (x^(n+1))/(n+1) + C, where C is the constant of integration. This constant is super important because the derivative of any constant is zero, so when we take the integral, we always need to remember to include that + C. Cool? Cool.

Let's also quickly refresh our memory on the rules of exponents. We'll need to rewrite sqrt(4x^7) in a way that’s easier to integrate. Recall that the square root is the same as raising something to the power of 1/2. And when we have multiple terms inside the square root, we can simplify them using the product rule of exponents. Knowing these rules will be crucial as we start simplifying our function.

Now, the main keyword here is integral of 8 * sqrt(4x^7) dx. This might seem like a complex problem, but it can be simplified into something much more manageable. The initial step is to break down the integral by simplifying sqrt(4x^7). This is the core of our problem, so let's get started. We'll need to rewrite the function so that we can apply our knowledge of the power rule and integration. Remember, the goal is to transform the function into a form that's easier to integrate. Keep these core concepts in mind as we progress, and you'll do great! We are going to address the formula and its solution step-by-step to arrive at the correct answer.

Step-by-Step Solution: Breaking Down the Integral

Okay, let's get our hands dirty and start solving this integral. The integral is ∫ 8 * sqrt(4x^7) dx. First, we can simplify the expression inside the square root. We can rewrite sqrt(4x^7) as sqrt(4) * sqrt(x^7). Because the square root of 4 is 2, we now have 2 * sqrt(x^7).

So, our integral now looks like this: ∫ 8 * 2 * sqrt(x^7) dx. Multiply the constants 8 and 2, and we get 16 * sqrt(x^7) dx. The next step is to rewrite the square root as a power of 1/2. Therefore, sqrt(x^7) becomes x^(7/2). Now our integral is ∫ 16 * x^(7/2) dx. See, things are getting simpler already! This is where the power rule of integration comes in handy.

To integrate 16 * x^(7/2) dx, we'll use the power rule. Recall the power rule: ∫ x^n dx = (x^(n+1))/(n+1) + C. In our case, n = 7/2. So, we add 1 to the exponent (7/2 + 1 = 9/2) and divide by the new exponent.

Let's apply this to our integral: ∫ 16 * x^(7/2) dx = 16 * (x^(7/2 + 1))/(7/2 + 1) + C. Simplify this to 16 * (x^(9/2))/(9/2) + C. We can further simplify this by dividing 16 by 9/2, which is the same as multiplying 16 by 2/9. So, 16 * (x^(9/2))/(9/2) = (32/9) * x^(9/2) + C. And there you have it, folks! The integral of 8 * sqrt(4x^7) dx is (32/9) * x^(9/2) + C.

So, to recap the steps: First, we simplified the initial expression. Next, we used the square root property and transformed the initial function to make it easier to solve using the power rule. Then, we applied the power rule for integration, and finally, we simplified the expression to obtain our final answer. It may seem like a long process, but once you break it down into manageable steps, it becomes easy to solve. The constant of integration, C, is important because it represents all the possible antiderivatives of our function. And with that, we're all done.

Simplifying the Expression: Making it Easier

Alright, let’s talk about simplifying the integral further. As we saw, the initial integral ∫ 8 * sqrt(4x^7) dx can be simplified to ∫ 16x^(7/2) dx. Now, how do we make sure our simplified expression is accurate and easy to work with? Well, we focus on breaking down the square root and applying the power rule effectively.

The key to simplifying this expression lies in understanding the properties of square roots and exponents. First, recognize that sqrt(4x^7) can be split into sqrt(4) * sqrt(x^7). This is a crucial step, and often overlooked by most, but is really essential to get to the solution. The sqrt(4) is simply 2, so you're left with 2 * sqrt(x^7). Next, you have to rewrite sqrt(x^7) as x^(7/2). This step is about understanding fractional exponents. Now you've got 2 * x^(7/2). Multiply that by the original 8 to get 16x^(7/2). This step-by-step simplification makes the integral much easier to tackle.

When simplifying the integral, always remember the order of operations. First, address any constants, then apply the rules of exponents, and finally, apply the integration rules. By simplifying step-by-step, you prevent errors and ensure that you get the correct answer. Simplification is not just about making the problem look neater; it's about transforming the problem into a form that's easier to solve using the known rules of calculus. Keeping things in order will save you tons of time. Simplification helps break down the complex parts of the equation into small parts that are easy to manage and easy to manipulate.

Applying the Power Rule: The Core of Integration

Okay, let's dive into how to apply the power rule in this context. The power rule is the core principle we use to integrate functions in the form of x^n. Remember, the power rule states that the integral of x^n dx is (x^(n+1))/(n+1) + C, where 'C' is the constant of integration. So when you have the simplified form 16x^(7/2) dx, it's time to put this rule into action.

With our simplified integral ∫ 16 * x^(7/2) dx, the first thing we do is add 1 to the exponent. So, 7/2 + 1 becomes 9/2. Then, divide the coefficient, which is 16, by the new exponent, which is 9/2. Dividing by a fraction is the same as multiplying by its reciprocal. So, it's 16 * (2/9). This gives us 32/9. Thus, the integral of 16 * x^(7/2) dx is (32/9) * x^(9/2) + C. The constant 'C' is essential, remember? It represents the constant of integration, reflecting the fact that the derivative of a constant is always zero.

Applying the power rule is like following a recipe. Know the formula, understand the step-by-step process, and then apply it. In our case, the crucial steps are: Identify the exponent (7/2), Add 1 to it (resulting in 9/2), Divide the coefficient (16) by the new exponent (9/2, or multiplying by 2/9), and don't forget the constant of integration (+ C). By breaking the process down this way, it's easier to apply the power rule effectively.

The Final Answer: Putting it All Together

Alright, guys, let's wrap it up and arrive at our final answer. Remember, our initial integral was ∫ 8 * sqrt(4x^7) dx. After simplifying and applying the power rule, we found the solution to be (32/9) * x^(9/2) + C. The x^(9/2) can also be written as x^4.5. This is the complete antiderivative of the original function.

So, what does this final answer mean? Well, if you were to take the derivative of (32/9) * x^(9/2) + C, you would get back the original function, 8 * sqrt(4x^7). That’s the beauty of integration; it’s the reverse process of differentiation. The constant 'C' reminds us that there are infinitely many antiderivatives, each differing by a constant value. The result (32/9) * x^(9/2) + C represents a family of functions.

In conclusion, we successfully navigated the integral. By simplifying the expression and applying the power rule, we found our answer. Integrating isn't always easy, but with practice and a good understanding of the steps involved, you can solve any integral. The process might seem complex at first, but with clear steps, practice, and the power of patience, any integral can be understood. Well done, everyone! Now, keep practicing and enjoy the world of calculus!