Find the most general antiderivative capital 𝐹 of 𝑥 of the function lowercase 𝑓 of 𝑥 is equal to four 𝑥 minus two.
The question gives us a linear function lowercase 𝑓 of 𝑥. And it wants us to find the most general antiderivative of this function. We’ll call this capital 𝐹 of 𝑥. Remember, we call capital 𝐹 of 𝑥 an antiderivative of lowercase 𝑓 of 𝑥 if the derivative of capital 𝐹 of 𝑥 with respect to 𝑥 is equal to lowercase 𝑓 of 𝑥. And one thing to keep in mind is antiderivatives are not unique. If we add any constant to an antiderivative, since the derivative of that constant is equal to zero, this function will also be an antiderivative of our function.
So you’ll often be asked to find the most general antiderivative of a function. When we’re asked to do this, we add a constant to our answer. This is to represent all the possible antiderivatives of our function. So let’s start by finding an antiderivative of our function. We see that our function, lowercase 𝑓 of 𝑥, is the linear function, four 𝑥 minus two. And since we can differentiate term by term, we can just find an antiderivative of each term separately. So let’s start by finding an antiderivative of four 𝑥.
We actually know how to find the antiderivative of a general term, 𝑎 times 𝑥 to the 𝑛th power. This is 𝑎 times 𝑥 to the power of 𝑛 plus one divided by 𝑛 plus one. And in the general case, remember, we would add a constant of integration. This is because to differentiate 𝑎 times 𝑥 to the power of 𝑛 plus one all divided by 𝑛 plus one, we multiply by our exponent and then reduce our exponent by one. This would give us 𝑎 times 𝑥 to the power of 𝑛. We can do this to find an antiderivative of four times 𝑥. We’ll write this as four times 𝑥 to the first power.
So to find an antiderivative of this expression, we add one to our exponent of one and then divide by this new exponent. This gives us four 𝑥 to the power of one plus one divided by one plus one. Of course, we can simplify one plus one to give us two. And then four divided by two is just equal to two. So one antiderivative of four 𝑥 is two 𝑥 squared. And sometimes, it might be worth thinking, “What happens if we differentiate this term?” We see if we differentiate this, we get four 𝑥. So this is indeed an antiderivative of four 𝑥.
Now, we just need to find an antiderivative of the constant negative two. To do this, remember, for any constant 𝑘, the derivative of 𝑘𝑥 with respect to 𝑥 is just equal to 𝑘. So for any constant 𝑘, we have that 𝑘𝑥 is an antiderivative of 𝑘. So by using 𝑘 is equal to negative two, we can see that an antiderivative of negative two is just negative two 𝑥.
So we found one possible antiderivative of our function, lowercase 𝑓 of 𝑥. But remember, this will be an antiderivative if we add any constant to this function. So we add a constant, we’ll call 𝐶, to make this the most general antiderivative of our function.
Therefore, we were able to show the most general antiderivative capital 𝐹 of 𝑥 of the function lowercase 𝑓 of 𝑥 is equal to four 𝑥 minus two is given by capital 𝐹 of 𝑥 is equal to two 𝑥 squared minus two 𝑥 plus 𝐶.