Video Transcript
Find the most general
antiderivative capital 𝐹 of 𝑥 of the function lower case 𝑓 of 𝑥 equals two 𝑥 to
the seventh power minus three 𝑥 to the fifth power minus 𝑥 squared.
To do this, we’ll take the
antiderivative of each of these terms separately. We need the antiderivative of two
times 𝑥 to the seventh power. We need to know what function, when
we take the derivative, equals two times 𝑥 to the seventh power. And for 𝑥 to the 𝑎 power, we can
find the antiderivative by taking 𝑥 to the 𝑎 plus one power over 𝑎 plus one. And then, in the general form, we
add 𝑐 to represent any constant. Let’s apply this to two times 𝑥 to
the seventh power. We’ll leave two to the side, and
we’ll take 𝑥 to the seven plus one power and divide by seven plus one.
The antiderivative of two times 𝑥
to the seventh power is two times 𝑥 to the eighth power over eight. And we can reduce this to one over
four. Two times 𝑥 to the seventh power
has an antiderivative of 𝑥 to the eighth power over four. And we’ll repeat this process with
negative three 𝑥 to the fifth. We can keep the negative three and
we have 𝑥 to the five plus one power all over five plus one. Negative three times 𝑥 to the
sixth power over six, which will reduce to 𝑥 to the sixth power over two. And we’ll make sure that we keep
that negative.
We’ll repeat the process one final
time. We’re dealing with negative 𝑥
squared, so I’ll pull out a negative one. Then we’ll have negative one times
𝑥 to the two plus one power over two plus one, negative 𝑥 cubed over three. Because we’re looking for the most
general form, we can’t forget this constant at the end. Which makes our antiderivative 𝐹
of 𝑥 equals 𝑥 to the eighth power over four minus 𝑥 to the sixth power over two
minus 𝑥 cubed over three plus 𝑐.
