Calculating The Integral Of X(2+x^5) Dx A Step-by-Step Guide

Hey guys! Today, let's dive into a fun calculus problem: calculating the integral of x(2+x^5) dx. Integrals might seem intimidating at first, but breaking them down step-by-step makes them much easier to handle. We're going to walk through this one together, making sure every step is crystal clear. So, grab your pencils, and let’s get started!

Understanding the Basics of Integration

Before we jump into the problem, let's quickly recap what integration is all about. In simple terms, integration is the reverse process of differentiation. While differentiation helps us find the rate of change of a function, integration helps us find the area under a curve. Think of it as piecing together the original function from its derivative. When you see the integral symbol ∫, it's asking you to find the antiderivative of the function that follows.

The fundamental theorem of calculus is our guiding star here. It connects differentiation and integration beautifully. The theorem states that if you have a function F(x) whose derivative is f(x), then the integral of f(x) is F(x) + C, where C is the constant of integration. That '+ C' is crucial because when we differentiate a constant, it disappears, so we need to account for all possible constants when we integrate. So, our mission is to find a function whose derivative matches the expression inside the integral.

Now, let's talk about some common integration rules. These are the tools in our calculus toolkit. One of the most frequently used rules is the power rule for integration. It says that the integral of x^n (where n is any number except -1) is (x^(n+1))/(n+1) + C. This rule is super handy for integrating polynomials, which are expressions with terms like x^2, x^5, and so on. Another important rule is the constant multiple rule, which states that the integral of a constant times a function is the constant times the integral of the function. For example, the integral of 5x^2 is 5 times the integral of x^2. Lastly, the sum/difference rule tells us that the integral of a sum or difference of functions is the sum or difference of their individual integrals. This means we can break down complex integrals into simpler ones. Knowing these basics will make the integral of x(2+x^5) dx much less daunting. So, keep these rules in mind as we move forward. We're going to apply them step-by-step to solve our integral problem, making sure you understand exactly how each rule comes into play. Ready to dive in? Let's go!

Step 1: Expanding the Expression

The first step in solving the integral of x(2+x^5) dx is to simplify the expression inside the integral. This makes the integration process much more manageable. We're going to use the distributive property to expand the expression. The distributive property, as you might recall, says that a(b+c) = ab + ac. So, we multiply the x outside the parentheses by each term inside the parentheses.

Here’s how it looks:

x(2 + x^5) = x * 2 + x * x^5

Now, let's simplify each term. x * 2 is simply 2x. For x * x^5, we use the rule of exponents that says x^a * x^b = x^(a+b). In this case, x is the same as x^1, so we have x^1 * x^5 = x^(1+5) = x^6. Putting it all together, our expanded expression is:

2x + x^6

So, instead of integrating x(2+x^5), we are now going to integrate 2x + x^6. This might seem like a small change, but it's a significant simplification. Expanding the expression allows us to deal with individual terms rather than a product of terms. This makes applying the integration rules much more straightforward. This step is crucial because it transforms a potentially complicated integral into a sum of simpler integrals, which we can easily handle using the power rule and the sum rule. Remember, the goal is to make the integral as easy as possible to solve, and expansion is a key technique to achieve that. By expanding the expression, we've set ourselves up for success in the next steps. Now, we're ready to apply the integration rules we talked about earlier. We’ll tackle each term separately and then combine the results. Keep this expanded form in mind as we move forward because it's the foundation for the rest of our solution. You'll see how much easier it becomes to integrate once you've expanded the expression. So, with our expanded expression in hand, let's proceed to the next step and start integrating each term. You're doing great so far! Keep up the excellent work, and we'll have this integral solved in no time.

Step 2: Applying the Power Rule of Integration

Now that we've expanded our expression to 2x + x^6, it's time to roll up our sleeves and apply the power rule of integration. Remember, the power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C, where n is any number except -1, and C is the constant of integration. We're going to apply this rule to each term in our expanded expression separately. This is where the fun really begins!

Let's start with the first term, 2x. We can rewrite 2x as 2x^1. Here, our n is 1. So, applying the power rule, we get:

∫2x^1 dx = 2 * ∫x^1 dx = 2 * (x^(1+1))/(1+1) + C

Simplifying this, we have:

2 * (x^2)/2 + C = x^2 + C

So, the integral of 2x is x^2 + C. Great! One term down, one to go. Now, let's tackle the second term, x^6. Here, our n is 6. Applying the power rule again, we get:

∫x^6 dx = (x^(6+1))/(6+1) + C

Simplifying this, we have:

(x^7)/7 + C

So, the integral of x^6 is (x^7)/7 + C. Awesome! We've integrated both terms separately using the power rule. You can see how straightforward the power rule makes integration when you have polynomial terms. Each term gets its exponent increased by one, and then we divide by the new exponent. Don't forget the '+ C' at the end of each integration – it's a crucial part of the solution because it represents all possible constant terms that could have disappeared during differentiation. Applying the power rule might seem like a simple step, but it's the heart of integrating polynomial expressions. It's a fundamental technique that you'll use over and over again in calculus. Now, we're just one step away from the final solution. We've integrated each term separately, and all that's left is to combine them. So, let's move on to the final step and put all the pieces together. You're doing fantastic! Keep going, and we'll have this integral conquered in no time.

Step 3: Combining the Results and Adding the Constant of Integration

Alright, guys, we're in the home stretch! We've successfully integrated each term of our expanded expression, and now it’s time to combine the results and add the all-important constant of integration. This is where everything comes together, and we see the final form of our integral. Remember, the constant of integration, C, is a crucial part of indefinite integrals because it accounts for any constant term that could have disappeared during differentiation. So, we can't forget about it!

In the previous step, we found that:

∫2x dx = x^2 + C

and

∫x^6 dx = (x^7)/7 + C

Now, we simply add these results together. When combining the results, we technically have a '+ C' for each integral, but since C represents any constant, we can combine all the constants into a single '+ C' at the end. It's like saying we have one big bucket for all the possible constant terms. So, our combined integral is:

x^2 + (x^7)/7 + C

And that's it! We've calculated the integral of x(2+x^5) dx. Pat yourselves on the back – you've done a fantastic job! This final step is a reminder that integration is a process of piecing things back together. We took a complex expression, broke it down into simpler terms, integrated each term separately, and then combined the results. The constant of integration is the final touch that makes our solution complete. Understanding the constant of integration is key to mastering indefinite integrals. It's not just a formality; it's a recognition that there are infinitely many functions that could have the same derivative. The '+ C' captures all those possibilities. Now that we have our final answer, it's a great idea to double-check our work. One way to do this is by differentiating our result and seeing if we get back to the original expression. If we differentiate x^2 + (x^7)/7 + C, we should get 2x + x^6, which is exactly what we started with after expanding. This verification step is a fantastic way to build confidence in your integration skills. So, congratulations! You've successfully navigated this integral problem. You've seen how to expand expressions, apply the power rule, and combine results with the constant of integration. Keep practicing, and you'll become an integration pro in no time. You've got this!

Conclusion

So, there you have it! We've successfully calculated the integral of x(2+x^5) dx step-by-step. We started by expanding the expression, then applied the power rule to each term, and finally combined the results, remembering to add the constant of integration. This problem is a great example of how breaking down complex problems into smaller, manageable steps can make even calculus feel approachable. Remember, practice is key. The more integrals you solve, the more comfortable you'll become with the techniques and the faster you'll be able to solve them. Don't be afraid to make mistakes – they're a natural part of the learning process. Each mistake is an opportunity to learn something new and strengthen your understanding. Keep experimenting with different types of integrals, and you'll be amazed at how quickly your skills improve. Calculus is a fascinating subject with countless applications in science, engineering, and beyond. By mastering the basics, you're opening doors to a world of possibilities. So, keep up the great work, stay curious, and never stop exploring the world of mathematics. You've got the tools, the knowledge, and the determination to succeed. Happy integrating, guys! And remember, every integral you solve is a victory. Keep celebrating those victories, and you'll go far. Now, go out there and conquer those integrals! You've got this!