Antiderivatives - Online Tutor, Practice Problems & Exam Prep

Antiderivatives

One of the concepts we should be quite familiar with at this point of the course is the process for taking derivatives. We've seen what derivatives are, and we've learned various techniques to take derivatives. But what if I threw a curveball at you and instead asked you to find an antiderivative? Now right at first, this might sound totally scary and abstract, but it turns out that an antiderivative is just the reverse process for taking a derivative. So what we've already learned about derivatives, we need to work backwards or do the reverse process to find an antiderivative.

Now I'm going to walk you through some examples in this video, and you're going to want to pay close attention, because antiderivatives are going to be very important as we make progress in this course. So let's just get right into an example. Now let's say I have some function \( f(x) \), and it's the function \( x^2 + 10 \). Now if I had this function to begin with, we should know the techniques for being able to take the derivative. So we go from \( f(x) \) to \( f'(x) \).

And doing this, we could use the rules we know. So I know how to use the power rule on \( x^2 \), which is going to give me \( 2x \). Then the derivative of 10 or any constant is just going to give me 0. So that means that our derivative is going to be \( 2x \), and that's going to be the solution for the derivative. Now this is something we should be familiar with.

But now let's see if we can do this reverse process by taking what's called the antiderivative. So if I want to take the antiderivative, well, in this case, when we took the derivative, we went from \( f(x) \) to \( f'(x) \). But by taking the antiderivative, meaning I'm starting with this function, \( 2x \) being this function that I have, that means that I'm going to go backwards to find the function \( F(x) \). So \( F(x) \) is going to be our antiderivative. And if I want to work backwards, well, if taking the derivative of \( x^2 \) gave me \( 2x \), that means the antiderivative of \( 2x \) would just be \( x^2 \).

So this is how we could take the antiderivative. This would be the process for doing this. But one of the problems we run into when finding this antiderivative is what exactly goes right here. I mean, before, this was a 10. But if I'm starting with \( 2x \) and going backwards, how could I figure out what this constant is?

I mean, it could be a 100, it could be 10, it could be negative 50, it could be 0. We don't know. And the truth is, is that with the information we're given, we actually can't know what this constant is. So what we just do is say that it's plus \( c \). We just leave this as some unknown constant here because we know the derivative got rid of whatever constant was there.

We just don't know exactly what it is. So this would be the solution for our antiderivative if we were to start with \( 2x \) and go backwards. So this is the process for taking antiderivatives. But to really make sure we have this concept down, let's actually try some more examples where we have to find antiderivatives. So here, in this first example, example A, we're asked to find the antiderivative of this function, \( 3x^2 \).

Now we're trying to find the antiderivative, which is called \( F(x) \). But how exactly do we find this? Well, I need to think to myself, what derivative would I have taken to give me \( 3x^2 \)? Well, this kinda looks like a power rule here because notice we have this 3 out front and we have a reduced power right there. So looking at this, I think \( x^3 \) is the function we're looking for.

And I know the power rule here would have given us \( 3x^2 \), so that is going to be our antiderivative. But don't forget, you have to add a plus constant \( c \) here because we don't know what constant was there in the first place when we took the derivative. So this right here would be our antiderivative and the solution to example A. But now let's move to example B. Example B asks us to find the antiderivative of this function, 3.

Now notice we're dealing with a constant here. So I want to find the antiderivative. Well, what derivative would leave me with just a constant? Well, if I think about it, that would be \( 3x \). Because if I had \( 3x \) and I took the derivative here, that \( x \) would just come off, leaving me with 3.

So the solution for this antiderivative would be \( 3x \), but then I need to add plus the constant that we don't know. So this right here would be the antiderivative and the solution to example B. But now let's move to our last example, example C. We need to find the antiderivative of this function, \( f(x) = 0 \). Now when doing this example, I'm going to go ahead and move myself to the other side of the screen here so we can solve this.

So I'm trying to find this function, \( F(x) \). And what would I have taken the derivative of to give me 0? Well, if I think about it, the derivative of any constant would give me 0. So that means that the antiderivative of 0 would just be some constant. Now we don't know what that constant is specifically, but we know that it would give us a constant for sure.

So \( F(x) = c \) would be the solution to this example. So this is how you can take antiderivatives and how you can do these types of problems where you need to reverse your process for the derivative. Now you may find that this is a little bit tricky to kinda think outside the box and figure out what derivative was there before, so that way you can work backwards to find what the antiderivative is. But thankfully, there is a process for checking your answer. All you need to do is take the derivative of whatever solution you've got, and that will tell you whether or not your function is correct.

So if I take the derivative of \( F(x) \), that should give me whatever this original function was, \( f(x) \). So we'll see if the functions match. Now I can use the power rule here. The derivative of \( x^3 \) is going to give me \( 3x^2 \), and the derivative of any constant is just going to be 0. So the derivative we get is \( 3x^2 \), and notice how this matches the original function we had.

So we did this one correctly. Now let's check this function right here. Well, we saw that we got \( 3x+c \), so let's take the derivative. The derivative of \( 3x \), well, that \( x \) is just going to come off, leaving us with just 3. The derivative of any constant is 0, so we end up with just 3.

And this is the same function we started with, so this is also correct. So we've checked off A, we've checked off B, now let's check C. Well, we have \( F(x) = c \), so let's see what \( f(x) \) would be. Well, \( f(x) \) is going to be the derivative of the constant \( c \), which is just 0. The _derivative of any constant is 0_, and this is the function we started with, meaning we did example C correctly as well.

So as you can see, there's a nice way to be able to check what your derivatives are and to see if you did the antiderivative correctly. So this is the process for taking antiderivatives. Hope you found this video helpful, and let's try getting a bit more practice with this concept.

Find the antiderivative of the following function.

$f\left(x\right)=200$

• A. $F\left(x\right)=0$
• B. $F\left(x\right)=200+C$
• C. $F\left(x\right)=200x$
• D. $F\left(x\right)=200x+C$

Find the antiderivative of the following function.

$f\left(x\right)=5x^4$

• A. $F\left(x\right)=25x^5+C$
• B. $F\left(x\right)=x^5+C$
• C. $F\left(x\right)=5x^5+C$
• D. $F\left(x\right)=\frac54x^5+C$

Find the antiderivative of the following function.

$f\left(x\right)=10x^9$

• A. $F\left(x\right)=x^{10}+C$
• B. $F\left(x\right)=10x^{10}+C$
• C. $F\left(x\right)=100x^{10}+C$
• D. $F\left(x\right)=\frac{10}{9}x^{10}+C$

Finding a Particular Antiderivative

In the last video, we got introduced to how we could find an antiderivative, and recall that this was just the reverse process of taking a derivative. Now in this video, we're going to learn how we can find particular antiderivatives. Recall, when we take the antiderivative of some function f(x), we would get the antiderivative F(x) plus some constant c. But we didn't really know how to find that constant or what it was. Well, in this video, we're going to see some problems where we actually need to figure out what this constant c is. I'm going to walk you through some examples of how we can do that, so let's just get right into things.

So let's say we have this example down here. We're asked to find the antiderivative of this function f(x) is equal to 2x. Now let's start with what we already know. We know how we can take the antiderivative of these functions by thinking about what would we need to have taken the derivative of to get this function 2x. Well, if I think about it, the function that works is x², because using the power rule, I can get 2x.

So if I took the derivative of this function, that would give me 2x, meaning that the antiderivative of 2x is x². But remember that you need to have plus some constant c because there was a possible constant that you took the derivative of that got rid of it when you got this answer. So you need to have x² plus c as your result, and that's the solution that we're familiar with. Now it turns out this actually has a name. It's the general antiderivative.

And the general antiderivative does not represent one particular solution or one antiderivative but rather a family of antiderivatives. Because since we have this constant c, this constant c represents any kind of vertical shift on our graph. So for example, this c here could be plus 1, meaning our function would be x² plus 1. We could also have a situation where the c value was negative 1. You would have x² minus 1.

This could also be something like negative 3, meaning it's x² minus 3. So if you think about it, we have an infinite family of possible functions this could be if we just leave it as x² plus c. But in this specific example, notice how we're actually given a point. We're given that the f(2) is equal to 3. So this is basically saying that for our function f(x), when x is equal to 2, our output y is equal to 3.

And what you need to do is you need to solve for c, which will give us the particular antiderivative that we're looking for. So going back down to this example here, well, we can see f(2) is equal to 3. So when x is equal to 2, we'll just plug 2 in for x there. We'll get 2 squared plus c. Our output is going to be 3.

So f(2) is going to be equal to 3. 2 squared here is 4, and then we have plus c. So if I want to solve for c, well, I just need to subtract this 4 on both sides of the equation. That's going to get the 4s to cancel on the right side, leaving me with c is equal to 3 minus 4. And 3 minus 4 is negative 1.

So this is our solution to c. So what that means is that the final equation that we have is going to replace the c with negative one, meaning it's going to be x² minus 1. So this is actually our particular antiderivative, which we can see right there. So this is what we call the particular antiderivative since we have a value for this constant. And we go over here to our graph, notice that we don't actually have this infinite family of antiderivatives, but we have one specific function that is going to represent the antiderivative that we just calculated.

So this is how you can find particular antiderivatives. So it's the same process as finding the general antiderivative, we just have this extra step of having to plug in whatever point we're given. Now to make sure that we're confident in solving these types of problems, let's try one more example down here where we don't have this kind of nice graph to help us out. So here we're asked to find the antiderivative of this function, f(x) is equal to 3x². Now first, I'm going to find the general antiderivative, and this is the process we talked about in the previous video.

So if I think about what value, what function I would have needed to take the derivative of to get 3x², well, notice we have a 3 here. We have a reduced power up here that looks like a power rule. So if I had x³ as my original function, the power rule using the derivative would give us 3x². So this right here would be the function. But don't forget, you need to add a plus c whenever you take an antiderivative.

So this would be what's called the general antiderivative. But to find the particular antiderivative, I need the point that we're given in this problem. So notice this says that f(1) is equal to 5. So what I'm going to do is take 1 and plug it in for x. So we're gonna have that f(1) is equal to 1 cubed plus c.

Now f(1), we can see right here, is equal to 5. So that means 5 is going to equal 1 cubed, which is just 1, plus the constant c. So you can go ahead and subtract 1 on both sides of this equation. That's going to get the ones to cancel on the right side, leaving me with c is equal to 5 minus 1, which is 4. So we get a c value of 4, meaning I need to take the c and replace it with 4 in this equation.

So our final result for the antiderivative is going to be that big F(x) is equal to x³plus the constant c, which we said is 4. So this is our solution, and that is how you can find particular antiderivatives. Just to sum things up, it's the same process we learned about in the previous video, where you're just reversing the derivative and then adding your constant c, but we just have this extra step of plugging a point in to get what that c value is, and then that will give us the final function by just replacing the c with whatever that constant was that we discovered. So that is how you can solve these types of problems where you need to find particular antiderivatives, and you know you're dealing with a particular antiderivative whenever you have a point given to you as well. So that's the solution.

Hope you found this video helpful.

For the following function $f\left(x\right)$, find the antiderivative $F\left(x\right)$ that satisfies the given condition.

$f\left(x\right)=5x^4$; $F\left(0\right)=1$

• A. $F\left(x\right)=5x^5+1$
• B. $F\left(x\right)=x^5+1$
• C. $F\left(x\right)=25x^5+1$
• D. $F\left(x\right)=x^5$

For the following function $f\left(x\right)$, find the antiderivative $F\left(x\right)$ that satisfies the given condition.

$f\left(x\right)=100x^{99}$; $F\left(1\right)=101$

• A. $F\left(x\right)=x^{100}+101$
• B. $F\left(x\right)=x^{100}+100$
• C. $F\left(x\right)=100x^{100}+101$
• D. $F\left(x\right)=100x^{100}+1$