### Video Transcript

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

Before we do anything else, weโll
calculate the general antiderivative. And that means weโll follow the
same process from the previous example. Weโll pull out the constant, add
one to our exponent, and then divide by the value of the new exponent. In this case, weโll have five times
๐ฅ to the fifth power divided by five. And weโll reduce that to ๐ฅ to the
fifth. Now, for the second term, take out
that four, weโll raise ๐ฅ cubed to ๐ฅ to the fourth power, and then divide by
four. Which reduces to ๐ฅ 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 ๐น of ๐ฅ
equals ๐ฅ to the fifth power plus ๐ฅ to the fourth power plus ๐. And we wanna plug in ๐น of one to
help us find the value of ๐. ๐น 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. Weโll take that information and
plug it in to what we found for the general antiderivative. An antiderivative under these
conditions is ๐ฅ to the fifth plus ๐ฅ to the fourth minus four.