### Video Transcript

Find the most general
antiderivative capital πΉ of π₯ of the function lower case π of π₯ equals two π₯ to
the seventh power minus three π₯ to the fifth power minus π₯ squared.

To do this, weβll take the
antiderivative of each of these terms separately. We need the antiderivative of two
times π₯ to the seventh power. We need to know what function, when
we take the derivative, equals two times π₯ to the seventh power. And for π₯ to the π power, we can
find the antiderivative by taking π₯ to the π plus one power over π plus one. And then, in the general form, we
add π to represent any constant. Letβs apply this to two times π₯ to
the seventh power. Weβll leave two to the side, and
weβll take π₯ to the seven plus one power and divide by seven plus one.

The antiderivative of two times π₯
to the seventh power is two times π₯ to the eighth power over eight. And we can reduce this to one over
four. Two times π₯ to the seventh power
has an antiderivative of π₯ to the eighth power over four. And weβll repeat this process with
negative three π₯ to the fifth. We can keep the negative three and
we have π₯ to the five plus one power all over five plus one. Negative three times π₯ to the
sixth power over six, which will reduce to π₯ to the sixth power over two. And weβll make sure that we keep
that negative.

Weβll repeat the process one final
time. Weβre dealing with negative π₯
squared, so Iβll pull out a negative one. Then weβll have negative one times
π₯ to the two plus one power over two plus one, negative π₯ cubed over three. Because weβre looking for the most
general form, we canβt forget this constant at the end. Which makes our antiderivative πΉ
of π₯ equals π₯ to the eighth power over four minus π₯ to the sixth power over two
minus π₯ cubed over three plus π.