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Question Video: Finding the General Antiderivative of a Polynomial Function Mathematics • 12th Grade

Find the most general antiderivative 𝐹(𝑥) of the function 𝑓(𝑥) = 2𝑥⁷ − 3𝑥⁵ − 𝑥².

02:34

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.

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