# Video: Finding the Antiderivative of a Polynomial Function

Determine the antiderivative 𝐹 of the function 𝑓(𝑥) = 5𝑥⁴ + 4𝑥³ where 𝐹(1) = −2.

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### Video Transcript

Determine the antiderivative capital 𝐹 of the function lower case 𝑓 of 𝑥 equals five 𝑥 to the fourth plus four 𝑥 cubed where capital 𝐹 of one equals negative two.

Before we do anything else, we’ll calculate the general antiderivative. And that means we’ll follow the same process from the previous example. We’ll pull out the constant, add one to our exponent, and then divide by the value of the new exponent. In this case, we’ll have five times 𝑥 to the fifth power divided by five. And we’ll reduce that to 𝑥 to the fifth. Now, for the second term, take out that four, we’ll raise 𝑥 cubed to 𝑥 to the fourth power, and then divide by four. Which reduces to 𝑥 to the fourth power. The four in the numerator and the denominator cancel out.

If we were finding the general form, we would add a constant 𝑐. And we say that capital 𝐹 of 𝑥 equals 𝑥 to the fifth power plus 𝑥 to the fourth power plus 𝑐. And we wanna plug in 𝐹 of one to help us find the value of 𝑐. 𝐹 of one equals negative two. One to the fifth power plus one to the fourth power. One plus one equals two. So two plus 𝑐 has to equal negative two. Subtract two from both sides. And we see that the constant value is negative four. We’ll take that information and plug it in to what we found for the general antiderivative. An antiderivative under these conditions is 𝑥 to the fifth plus 𝑥 to the fourth minus four.