Finding The General Antiderivative Of F(x) = 7x^5 + 9/x^6 + 7

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Finding the General Antiderivative of f(x) = 7x^5 + 9/x^6 + 7

Hey guys! Today, we're diving into a common calculus problem: finding the most general antiderivative of a function. Specifically, we're going to tackle the function f(x) = 7x^5 + 9/x^6 + 7. This might seem a bit daunting at first, but don't worry, we'll break it down step-by-step so it's super easy to understand. Think of antiderivatives as the reverse process of taking derivatives. It's like retracing your steps – if taking the derivative is going forward, finding the antiderivative is going backward. So, let's put on our detective hats and get started!

Understanding Antiderivatives

Before we jump into the problem, let's quickly recap what antiderivatives are all about. In essence, the antiderivative of a function f(x) is another function F(x) whose derivative is f(x). Mathematically, we express this as F'(x) = f(x). Now, here's a crucial point: antiderivatives aren't unique. 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 (where C is any constant) is also an antiderivative. This is why we talk about the "most general antiderivative" – it includes that arbitrary constant C, acknowledging the infinite possibilities. This constant, C, is often called the constant of integration. It's like the missing piece of the puzzle when we're going backward from the derivative to the original function. Imagine you're trying to figure out a person's exact age based on their growth rate; you'd need some initial age to start from, right? C is that initial value in the world of calculus. So, when we're finding antiderivatives, we're not just looking for one function; we're looking for a whole family of functions that differ only by a constant. This is why you'll always see that "+ C" at the end of an antiderivative. It's the universal symbol for "there might be a constant here!"

The Power Rule for Integration

To find antiderivatives, we often use the power rule for integration, which is essentially the reverse of the power rule for differentiation. The power rule for integration states that the antiderivative of x^n (where n is any real number except -1) is (x^(n+1))/(n+1) + C. Let’s unpack this a bit. The rule says, you increase the exponent by one (n+1), then divide by the new exponent (n+1). And, of course, we always add our constant of integration, C. It's crucial to remember that this rule doesn't apply when n = -1, because we'd be dividing by zero. The antiderivative of x^(-1), or 1/x, is a special case that involves the natural logarithm, which we'll touch on later. But for most polynomial terms, the power rule is our go-to tool. Think of it as the workhorse of integration. Now, why does this rule work? Well, it's the reverse of the power rule for differentiation. When we differentiate x^n, we multiply by n and decrease the exponent by one. Integration does the opposite: we increase the exponent by one and divide by the new exponent. They're perfect inverses, like addition and subtraction or multiplication and division. To illustrate, consider x^2. According to the power rule, its antiderivative is (x^(2+1))/(2+1) + C = (x^3)/3 + C. If you differentiate (x^3)/3 + C, you'll indeed get back to x^2. See? It's like magic, but it's actually just math!

Applying the Power Rule to Our Function

Now, let’s apply this powerful rule to our function, f(x) = 7x^5 + 9/x^6 + 7. First, it's super helpful to rewrite the function to make the exponents clearer. We can rewrite 9/x^6 as 9x^(-6). This makes it much easier to apply the power rule. Our function now looks like this: f(x) = 7x^5 + 9x^(-6) + 7. Remember, we're dealing with three terms here, and we can find the antiderivative of each term separately, thanks to the linearity of integration (the antiderivative of a sum is the sum of the antiderivatives). Let's tackle each term one by one. For the first term, 7x^5, we apply the power rule: increase the exponent by 1 (5+1 = 6), divide by the new exponent (6), and keep the constant multiplier (7). This gives us (7x^6)/6. Next, let's handle the second term, 9x^(-6). Again, we apply the power rule: increase the exponent by 1 (-6+1 = -5), divide by the new exponent (-5), and keep the constant multiplier (9). This gives us (9x^(-5))/(-5), which can be simplified to -9/(5x^5). Finally, the third term is just a constant, 7. Remember, the antiderivative of a constant k is kx + C. So, the antiderivative of 7 is 7x. Now we have the antiderivatives of all three terms. All that’s left to do is put them together and add our constant of integration, C. This C is super important, guys! Don't forget it!

Putting It All Together

Okay, we've found the antiderivatives of each term individually. Now it's time to assemble the final answer. We found that the antiderivative of 7x^5 is (7x^6)/6, the antiderivative of 9x^(-6) is -9/(5x^5), and the antiderivative of 7 is 7x. So, we just add them all up and tack on our trusty "+ C" at the end. This gives us the most general antiderivative: F(x) = (7x^6)/6 - 9/(5x^5) + 7x + C. And there you have it! That's the most general antiderivative of our original function, f(x) = 7x^5 + 9/x^6 + 7. It might look a little complicated, but remember, we built it up step-by-step using the power rule and the understanding that antiderivatives aren't unique – that's why we have the C. You can always check your work by differentiating F(x). If you've done everything correctly, you should get back to the original function, f(x). It's like a built-in self-check! This is a fantastic way to confirm that you haven't made any mistakes along the way. Integration can be tricky, but with practice and a solid understanding of the rules, you'll be finding antiderivatives like a pro in no time!

Checking Our Work

It's always a good idea to check your work in calculus, especially when dealing with antiderivatives. A simple way to verify our answer is to differentiate the antiderivative we found, F(x) = (7x^6)/6 - 9/(5x^5) + 7x + C, and see if we get back our original function, f(x) = 7x^5 + 9/x^6 + 7. Let's go through the differentiation step by step. First, we differentiate (7x^6)/6. Using the power rule for differentiation, we multiply by the exponent (6) and decrease the exponent by 1 (6-1 = 5). The 6 in the numerator and denominator cancel out, leaving us with 7x^5. Next, we differentiate -9/(5x^5), which can be rewritten as (-9/5)x^(-5). Again, using the power rule, we multiply by the exponent (-5) and decrease the exponent by 1 (-5-1 = -6). This gives us (-9/5) * (-5) * x^(-6), which simplifies to 9x^(-6), or 9/x^6. Then, we differentiate 7x, which simply gives us 7. Finally, the derivative of the constant C is 0. Now, let's put it all together: the derivative of F(x) is 7x^5 + 9/x^6 + 7 + 0, which is exactly our original function, f(x). Yay! This confirms that our antiderivative is correct. Checking your work not only gives you confidence in your answer but also reinforces your understanding of the concepts. It's a valuable habit to develop in calculus and beyond. So, always take a few moments to check your solutions whenever possible. It can save you from making careless errors and help solidify your knowledge.

Conclusion

So, there you have it! We've successfully found the most general antiderivative of the function f(x) = 7x^5 + 9/x^6 + 7. We started by understanding the concept of antiderivatives and the importance of the constant of integration, C. Then, we reviewed the power rule for integration, a key tool for finding antiderivatives of polynomial terms. We applied the power rule to each term in our function, being careful to rewrite the expression with negative exponents where necessary. Finally, we combined the antiderivatives of each term and added the constant C to get the most general antiderivative: F(x) = (7x^6)/6 - 9/(5x^5) + 7x + C. And, to be extra sure, we even checked our work by differentiating F(x) and verifying that it matched our original function, f(x). Finding antiderivatives is a fundamental skill in calculus, and mastering it opens doors to many other concepts, such as definite integrals and differential equations. The more you practice, the more comfortable you'll become with these techniques. Remember, calculus is like building with blocks; each concept builds upon the previous one. So, keep practicing, keep asking questions, and you'll become a calculus whiz in no time! Keep up the great work, guys! You've got this! Remember, practice makes perfect, so keep at it, and you'll be a master of antiderivatives in no time!