Question Video: Finding the General Antiderivative of a Quadratic Function Mathematics

Video Transcript

Find the most general antiderivative capital 𝐹 of 𝑥 of the function lowercase 𝑓 of 𝑥 is equal to 𝑥 minus three all squared.

The question gives us a function lowercase 𝑓 of 𝑥. It wants us to find the most general antiderivative of this function. We’ll call this capital 𝐹 of 𝑥. Remember, an antiderivative means when we differentiate this, we get back to our original function. In other words, we want capital 𝐹 prime of 𝑥 to be equal to lowercase 𝑓 of 𝑥. And remember, since the derivative of any constant is equal to zero, we can add any constant we want to our antiderivative. And it will still be an antiderivative of our function lowercase 𝑓 of 𝑥.

So we add a constant 𝐶 to our antiderivative. We call this the most general antiderivative, since it will represent all antiderivatives of our function. So let’s start trying to find our antiderivative. Let’s start by looking at our function lowercase 𝑓 of 𝑥. We can see it’s 𝑥 minus three all squared. And this is a problem, since this is a composition of functions. This can make it much more difficult to find our antiderivatives. Instead, let’s simplify our function by distributing the square over our parentheses. We’ll distribute our square by using the FOIL method. We’ll start by multiplying the first term of each factor together. This gives us 𝑥 times 𝑥, which is equal to 𝑥 squared.

Next, the FOIL method tells us to multiply our two outer terms together. This gives us 𝑥 multiplied by negative three, which is negative three 𝑥. Now, we want to multiply our inner two terms together. Again, that’s negative three times 𝑥, which is negative three 𝑥. Finally, we want to multiply the last term of each factor together. This gives us negative three times negative three, which is equal to nine. And we can simplify this, since negative three 𝑥 minus three 𝑥 is equal to negative six 𝑥.

So now, we can see our function lowercase 𝑓 of 𝑥 is a polynomial. And we know how to find the antiderivative of each term in a polynomial separately. We know to find the antiderivative of 𝑎 multiplied by 𝑥 to the 𝑛th power, we want to add one to our exponent of 𝑥 and then divide by this new exponent of 𝑥. This gives us 𝑎 times 𝑥 to the power of 𝑛 plus one divided by 𝑛 plus one. And in the general case, we’ll add a constant of integration 𝐶. We’ll want to do this to each term separately. Let’s start with 𝑥 squared. To find an antiderivative of 𝑥 squared, we can add one to our exponent of two. This gives us a new exponent of three. But remember, we then need to divide by this new exponent. This gives us 𝑥 cubed over three.

We now want to find an antiderivative of our second term, negative six 𝑥. One way of doing this is to rewrite our second term as negative six times 𝑥 to the first power. Again, we want to add one to our exponent of 𝑥. This time, it’s equal to one. So we get two and then we divide by two. So we have negative six 𝑥 squared divided by two. And we know six divided by two is equal to three. So we’ve shown negative three 𝑥 squared is an antiderivative of negative six 𝑥. Finally, we want to find an antiderivative of the third term, nine. We might be tempted to write this as nine times 𝑥 to the zeroth power.

However, there’s a more simple method of finding an antiderivative in this case. We know for any constant 𝐾, the derivative of 𝐾𝑥 with respect to 𝑥 is just equal to 𝐾. In other words, for any constant 𝐾, 𝐾𝑥 is an antiderivative of 𝐾. So when 𝐾 is equal to nine, we see that nine 𝑥 is an antiderivative of nine. So to find an antiderivative of any constant, we just need to multiply that constant by 𝑥.

Remember, though, this is just one possible antiderivative of our function lowercase 𝑓 of 𝑥. If we add any constant to this, the derivative of that constant is equal to zero. So this is still an antiderivative of our function. So we add a constant we’ve called 𝐶 to this function. This represents all of our possible antiderivatives. And this is how we find our most general antiderivative.

Therefore, we were able to show the most general antiderivative capital 𝐹 of 𝑥 of the function lowercase 𝑓 of 𝑥 is equal to 𝑥 minus three all squared is given by capital 𝐹 of 𝑥 is equal to 𝑥 cubed over three minus three 𝑥 squared plus nine 𝑥 plus a constant of integration 𝐶.