Video: Finding the Antiderivative of a Polynomial Function

Determine the antiderivative ๐น of the function ๐‘“(๐‘ฅ) = 5๐‘ฅโด + 4๐‘ฅยณ where ๐น(1) = โˆ’2.


Video Transcript

Determine the antiderivative capital ๐น of the function lower case ๐‘“ of ๐‘ฅ equals five ๐‘ฅ to the fourth plus four ๐‘ฅ cubed where capital ๐น of one equals negative two.

Before we do anything else, weโ€™ll calculate the general antiderivative. And that means weโ€™ll follow the same process from the previous example. Weโ€™ll pull out the constant, add one to our exponent, and then divide by the value of the new exponent. In this case, weโ€™ll have five times ๐‘ฅ to the fifth power divided by five. And weโ€™ll reduce that to ๐‘ฅ to the fifth. Now, for the second term, take out that four, weโ€™ll raise ๐‘ฅ cubed to ๐‘ฅ to the fourth power, and then divide by four. Which reduces to ๐‘ฅ 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 ๐‘. And we say that capital ๐น of ๐‘ฅ equals ๐‘ฅ to the fifth power plus ๐‘ฅ to the fourth power plus ๐‘. And we wanna plug in ๐น of one to help us find the value of ๐‘. ๐น of one equals negative two. One to the fifth power plus one to the fourth power. One plus one equals two. So two plus ๐‘ has to equal negative two. Subtract two from both sides. And we see that the constant value is negative four. Weโ€™ll take that information and plug it in to what we found for the general antiderivative. An antiderivative under these conditions is ๐‘ฅ to the fifth plus ๐‘ฅ to the fourth minus four.

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