Antiderivative Of 2e^x + 7: Step-by-Step Solution

Oct 31, 2025 by Admin 50 views

Hey guys! Today, we're diving into a fun little calculus problem: finding the most general antiderivative of the function $\frac{dy}{dx} = 2e^x + 7$ . If you're scratching your head thinking, "What's an antiderivative?" or "How do I even start?", don't worry! We're going to break it down step by step, so you'll be a pro in no time. This is a crucial concept in calculus, and mastering it opens doors to solving more complex problems involving integrals and differential equations. So, let's get started and unravel this antiderivative mystery together!

Understanding Antiderivatives

Before we jump into the nitty-gritty, let's make sure we're all on the same page about what an antiderivative actually is. In simple terms, an antiderivative is the reverse operation of a derivative. Think of it like this: if taking the derivative is like going from point A to point B, finding the antiderivative is like finding the path to go from point B back to point A. Mathematically, if we have a function f(x), its antiderivative, often denoted as F(x), is a function whose derivative is f(x). That is, F'(x) = f(x). The concept of antiderivatives is fundamental to integral calculus, serving as the backbone for understanding indefinite integrals and solving differential equations. It’s more than just reversing differentiation; it’s about understanding the family of functions that could have the given derivative.

The most crucial aspect to remember is the "most general" part. When we find an antiderivative, we're not just looking for one function; we're looking for a family of functions. Why? Because the derivative of a constant is always zero. This means that if F(x) is an antiderivative of f(x), then F(x) + C is also an antiderivative, where C is any constant. This C is called the constant of integration, and it's super important! For example, consider the function f(x) = 2x. One antiderivative is F(x) = x^2, since the derivative of x^2 is 2x. But x^2 + 1, x^2 - 5, and x^2 + 100 are also antiderivatives because their derivatives are also 2x. This is why we always add the constant of integration, C, to our antiderivative, representing the entire family of functions that could be the antiderivative.

So, in summary, finding the antiderivative is like undoing the derivative. We’re looking for the function (or, more accurately, the family of functions) whose derivative matches the given function. And remember that constant of integration, C! It's the key to capturing all possible antiderivatives. Understanding this concept is the first step in mastering integration, which is a cornerstone of calculus and essential for many applications in science, engineering, and economics. Let's move on and see how we can apply this understanding to solve our specific problem.

Breaking Down the Function dy/dx = 2e^x + 7

Okay, now that we've got a handle on what antiderivatives are all about, let's zero in on the function we need to tackle: $\frac{dy}{dx} = 2e^x + 7$ . This function might look a little intimidating at first, especially if you're just starting out with calculus. But don't sweat it! We're going to break it down into smaller, more manageable pieces. The key here is to recognize that we can find the antiderivative of each term separately, thanks to a handy rule called the sum rule for antiderivatives. This rule basically says that the antiderivative of a sum is the sum of the antiderivatives. This makes our task much easier, as we can focus on finding the antiderivative of $2e^x$ and the antiderivative of 7 independently, and then simply add them together. Isn't that neat?

So, let's look at each term individually. First, we have $2e^x$ . This term involves the exponential function $e^x$ , which has a very special property: its derivative is itself! That is, $\frac{d}{dx}(e^x) = e^x$ . This means that the antiderivative of $e^x$ is also $e^x$ ! Now, we have a constant multiplier of 2 in front of the $e^x$ . The rule for constant multiples tells us that we can simply carry this constant along when finding the antiderivative. In other words, the antiderivative of $2e^x$ is just 2 times the antiderivative of $e^x$ , which gives us $2e^x$ . Remember, calculus is all about recognizing patterns and using rules to simplify complex problems. By breaking down the function into smaller parts, we can apply these rules more effectively.

Next, we have the term 7. This is a constant function. To find its antiderivative, we need to think about what function, when differentiated, would give us 7. Recall that the power rule for differentiation states that the derivative of $x^n$ is $nx^{n-1}$ . Reversing this, we know that the antiderivative of a constant k is kx. Therefore, the antiderivative of 7 is 7x. It's like asking ourselves, "What function's slope is constantly 7?" The answer is a straight line with a slope of 7, which is represented by the equation 7x. By understanding the basic rules of differentiation and antidifferentiation, we can systematically approach these problems and find the solutions with confidence. Now that we've dissected the function and found the antiderivatives of its individual terms, we're ready to put it all together and find the most general antiderivative.

Finding the Antiderivative of 2e^x

Alright, let's zoom in on finding the antiderivative of $2e^x$ . This part is actually pretty straightforward once you remember the golden rule about the exponential function $e^x$ . As we touched on earlier, the derivative of $e^x$ is, well, $e^x$ itself! This makes $e^x$ a unique and incredibly useful function in calculus. So, naturally, the antiderivative of $e^x$ is also $e^x$ . It's like a mathematical mirror – what you put in is what you get out! But what about that 2 hanging out in front? Remember the constant multiple rule we mentioned? This rule is your best friend when dealing with constants multiplied by functions. It simply states that if you have a constant multiplied by a function, you can just carry the constant along when finding the antiderivative.

So, here's how it works: If the function is $2e^x$ , the antiderivative will be 2 times the antiderivative of $e^x$ . Since the antiderivative of $e^x$ is $e^x$ , the antiderivative of $2e^x$ is simply $2e^x$ . Easy peasy, right? This might seem like a small step, but it's a crucial one. Mastering these basic antiderivative rules is like building a strong foundation for a house – you need it to support the more complex structures you'll build later on. The constant multiple rule is one of those fundamental tools that you'll use over and over again in calculus, so make sure you're comfortable with it. It allows us to handle constants with ease, making the process of finding antiderivatives much smoother.

To solidify this concept, let’s think about why this works. When we take the derivative of $2e^x$ , we get $2e^x$ , confirming that $2e^x$ is indeed the antiderivative of $2e^x$ . This reinforces the idea that antidifferentiation is the reverse process of differentiation. Now that we've conquered the $2e^x$ term, let's move on to the next piece of the puzzle: finding the antiderivative of the constant term, 7. Remember, each step we take brings us closer to the complete solution, and by breaking the problem down into manageable parts, we make the whole process much less daunting. So, let’s keep going and see how to tackle that constant term.

Finding the Antiderivative of 7

Now, let's tackle the second part of our function: the constant term, 7. Finding the antiderivative of a constant might seem a bit too simple, but it's an important step, and it helps reinforce our understanding of antiderivatives. Remember, the antiderivative of a function is the function whose derivative gives us the original function. So, we need to ask ourselves: what function, when differentiated, gives us 7? Think back to the power rule for differentiation. The power rule states that the derivative of $x^n$ is $nx^{n-1}$ . To reverse this, we need to increase the power by 1 and divide by the new power. However, a simpler way to think about the antiderivative of a constant is to remember that the derivative of kx (where k is a constant) is just k. This is a direct application of the power rule when n=1.

In our case, the constant is 7. So, we're looking for a function whose derivative is 7. Following the rule, the antiderivative of 7 is simply 7x. Why? Because the derivative of 7x is indeed 7. It's like finding the slope of a line – a line with a constant slope of 7 is represented by the equation 7x. This connection between slopes and antiderivatives is a fundamental concept in calculus and helps us visualize what we're actually doing when we find an antiderivative. We're essentially finding a function whose rate of change (derivative) matches the given function.

This might seem almost too easy, but it's essential to understand the basics thoroughly. Constants are common in mathematical functions, and knowing how to handle them quickly and accurately will save you time and prevent errors. Now that we've found the antiderivative of both $2e^x$ and 7, we're ready to combine them and add the all-important constant of integration. This is where we'll see the full power of understanding antiderivatives and how they represent a family of functions, not just one single function. So, let's put it all together and find the most general antiderivative of our original function.

Combining the Antiderivatives and the Constant of Integration

We've done the heavy lifting! We've found the antiderivative of $2e^x$ , which is $2e^x$ , and we've found the antiderivative of 7, which is 7x. Now comes the really satisfying part: putting it all together! Remember the sum rule for antiderivatives? It says that the antiderivative of a sum is the sum of the antiderivatives. So, to find the antiderivative of $\frac{dy}{dx} = 2e^x + 7$ , we simply add the antiderivatives we found for each term.

This gives us $2e^x + 7x$ . But hold on! We're not quite done yet. There's one super important thing we need to remember: the constant of integration, C. As we discussed earlier, the derivative of a constant is always zero. This means that when we find an antiderivative, we're actually finding a family of functions, all differing by a constant. To represent this family, we add C to our antiderivative. It's like saying, "Hey, there could be any constant term here, so we need to account for it."

So, the most general antiderivative of $\frac{dy}{dx} = 2e^x + 7$ is $2e^x + 7x + C$ . And that's it! We've done it! We've successfully found the most general antiderivative of our function. This C is not just a formality; it's crucial for representing the infinite possibilities for the antiderivative. For instance, $2e^x + 7x + 1$ , $2e^x + 7x - 5$ , and $2e^x + 7x + 100$ are all valid antiderivatives of the given function. Without the C, we'd be missing out on this entire family of solutions.

Let's take a moment to appreciate what we've accomplished. We started with a function that might have seemed a bit intimidating, but by breaking it down into smaller parts, applying the rules of antiderivatives, and remembering the constant of integration, we were able to find the complete solution. This process highlights the power of calculus: turning complex problems into manageable steps. Now, let’s summarize our findings and reflect on the key takeaways from this problem.

Conclusion: The Most General Antiderivative

Alright, let's wrap things up! We've journeyed through the world of antiderivatives and successfully found the most general antiderivative of the function $\frac{dy}{dx} = 2e^x + 7$ . Our final answer, after all the calculations and considerations, is $2e^x + 7x + C$ . Remember, the key steps we took were:

Understanding Antiderivatives: We started by grasping the concept of antiderivatives as the reverse process of differentiation and the importance of the constant of integration.
Breaking Down the Function: We split the function into its individual terms, $2e^x$ and 7, to make the problem more manageable.
Finding Individual Antiderivatives: We found the antiderivative of $2e^x$ using the constant multiple rule and the fact that the antiderivative of $e^x$ is $e^x$ . Then, we found the antiderivative of 7 using the rule for the antiderivative of a constant.
Combining and Adding the Constant of Integration: We added the individual antiderivatives together and included the constant of integration, C, to represent the most general antiderivative.

This exercise not only provides the solution to a specific problem but also reinforces the fundamental principles of calculus. The constant of integration, C, is a crucial element that signifies that there isn't just one antiderivative, but an infinite number of them, differing only by a constant. This concept is vital for understanding indefinite integrals and their applications in various fields. By mastering the basics, like finding the antiderivative of simple functions and understanding the rules of integration, you build a solid foundation for tackling more complex calculus problems.

So, next time you encounter an antiderivative problem, remember to break it down, apply the rules, and don't forget that C! You've got this! Keep practicing, and you'll become a calculus whiz in no time. Happy calculating, guys! Remember, calculus isn't just about numbers and equations; it's about understanding the relationships and patterns that govern the world around us. So, embrace the challenge, enjoy the process, and keep exploring the fascinating world of mathematics!