Finding Antiderivatives Step-by-Step A Comprehensive Guide

Hey everyone! Today, we're diving into the exciting world of antiderivatives, a fundamental concept in calculus. We'll be focusing on finding the antiderivative for a few functions, specifically when the constant of integration, C, is equal to 0. But that's not all! We'll also be checking our answers by using differentiation, the inverse operation of finding antiderivatives. So, let's get started and make sure you understand each step along the way, guys!

Understanding Antiderivatives

Before we jump into the problems, let's quickly recap what antiderivatives are all about. In simple terms, an antiderivative of a function f(x) is another function F(x) whose derivative is f(x). Think of it as reversing the process of differentiation. When we find an antiderivative, we're essentially asking, "What function, when differentiated, gives us this original function?" The key is that the antiderivative isn't unique. We always have to add a constant of integration, C, because the derivative of a constant is zero. This means that F(x) + C is also an antiderivative of f(x). For this exercise, we're setting C to 0 to simplify things a bit.

Why Antiderivatives Matter

You might be wondering, why bother with antiderivatives anyway? Well, they're super important in calculus and have tons of applications in various fields. For instance, antiderivatives are crucial for calculating areas under curves (definite integrals), solving differential equations (which model many real-world phenomena), and even in physics for finding displacement from velocity or velocity from acceleration. Understanding antiderivatives is a building block for more advanced calculus concepts. So, paying attention to the details now will definitely pay off later. Plus, it's kind of like solving a puzzle – figuring out the original function from its derivative is a fun challenge!

The Power Rule in Reverse

One of the most useful tools for finding antiderivatives is the power rule, but we'll be using it in reverse. Remember the power rule for differentiation? It says that the derivative of xⁿ is nxⁿ⁻¹. To reverse this, we increase the exponent by 1 and then divide by the new exponent. So, the antiderivative of xⁿ (where n ≠ -1) is ( xⁿ⁺¹ ) / ( n+1 ) + C. This rule will be our best friend for the problems we're tackling today. Just remember the exception: when n = -1, the power rule doesn't apply, and we need to use the natural logarithm instead (but we won't encounter that case in these specific problems, thankfully!).

Problem (a): g(x) = 1/x²

Let's kick things off with our first function: g(x) = 1/x². Our mission is to find its antiderivative, G(x), with the condition that C = 0. The first step is to rewrite g(x) in a form that's easier to work with. We can express 1/x² as x⁻². Now, we can clearly see that this is a power function, and we can apply the reverse power rule. Remember, the reverse power rule is all about adding one to the exponent and dividing by the new exponent.

Applying the Reverse Power Rule

So, let's do it! We start with x⁻². Adding 1 to the exponent, we get -2 + 1 = -1. So, our new exponent is -1. Now, we divide by this new exponent: (x⁻¹) / (-1). This simplifies to - x⁻¹. And that, my friends, is the antiderivative! But wait, we're not quite done yet. We need to remember our constant of integration, C. However, in this case, we're told that C = 0, so we can just leave it out. Therefore, G(x) = -x⁻¹, which can also be written as G(x) = -1/x. That's our answer for the antiderivative.

Verifying with Differentiation

But how do we know we're right? This is where differentiation comes to the rescue! To check our answer, we'll differentiate G(x) = -1/x and see if we get back to our original function, g(x) = 1/x². Again, it's helpful to rewrite -1/x as - x⁻¹. Now, we can apply the regular power rule for differentiation. The power rule says that the derivative of xⁿ is nxⁿ⁻¹. So, the derivative of - x⁻¹ is -(-1) x⁻¹⁻¹ = x⁻². And guess what? That's exactly 1/x², which is our original function g(x)! We did it! By differentiating our antiderivative, we've confirmed that we found the correct answer. This step is so important because it gives us confidence in our result and helps catch any potential mistakes.

Problem (b): h(x) = 9/x²

Next up, we have h(x) = 9/x². This function looks quite similar to the previous one, but with an added constant factor of 9. This constant will play a role in our calculations, but don't worry, it's a straightforward process. Just like before, our goal is to find the antiderivative, H(x), with C = 0, and then verify our answer using differentiation. Remember the key is to keep track of the constant multiple.

Handling the Constant Multiple

The first thing we'll do is rewrite h(x) in a more convenient form. We can write 9/x² as 9 x⁻². Now, we have a constant (9) multiplied by a power function (x⁻²). When finding antiderivatives, we can simply carry the constant along. In other words, the antiderivative of 9 x⁻² will be 9 times the antiderivative of x⁻². This is a handy property that makes dealing with constants much easier. It's like saying,