Video: Finding the General Antiderivative of a Polynomial Function

Find the most general antiderivative 𝐹(π‘₯) of the function 𝑓(π‘₯) = 2π‘₯⁷ βˆ’ 3π‘₯⁡ βˆ’ π‘₯Β².

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Video Transcript

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

To do this, we’ll 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‘Ž power, we can find the antiderivative by taking π‘₯ to the π‘Ž plus one power over π‘Ž plus one. And then, in the general form, we add 𝑐 to represent any constant. Let’s apply this to two times π‘₯ to the seventh power. We’ll leave two to the side, and we’ll 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 we’ll 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 we’ll make sure that we keep that negative.

We’ll repeat the process one final time. We’re dealing with negative π‘₯ squared, so I’ll pull out a negative one. Then we’ll have negative one times π‘₯ to the two plus one power over two plus one, negative π‘₯ 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 π‘₯ equals π‘₯ to the eighth power over four minus π‘₯ to the sixth power over two minus π‘₯ cubed over three plus 𝑐.

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