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

Find the most general antiderivative ๐น(๐ฅ) of the function ๐(๐ฅ) = 2๐ฅโท โ 3๐ฅโต โ ๐ฅยฒ.

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### Video Transcript

Find the most general antiderivative capital ๐น of ๐ฅ of the function lower case ๐ of ๐ฅ equals two ๐ฅ to the seventh power minus three ๐ฅ to the fifth power minus ๐ฅ squared.

To do this, weโll take the antiderivative of each of these terms separately. We need the antiderivative of two times ๐ฅ to the seventh power. We need to know what function, when we take the derivative, equals two times ๐ฅ to the seventh power. And for ๐ฅ to the ๐ power, we can find the antiderivative by taking ๐ฅ to the ๐ plus one power over ๐ plus one. And then, in the general form, we add ๐ to represent any constant. Letโs apply this to two times ๐ฅ to the seventh power. Weโll leave two to the side, and weโll take ๐ฅ to the seven plus one power and divide by seven plus one.

The antiderivative of two times ๐ฅ to the seventh power is two times ๐ฅ to the eighth power over eight. And we can reduce this to one over four. Two times ๐ฅ to the seventh power has an antiderivative of ๐ฅ to the eighth power over four. And weโll repeat this process with negative three ๐ฅ to the fifth. We can keep the negative three and we have ๐ฅ to the five plus one power all over five plus one. Negative three times ๐ฅ to the sixth power over six, which will reduce to ๐ฅ to the sixth power over two. And weโll make sure that we keep that negative.

Weโll repeat the process one final time. Weโre dealing with negative ๐ฅ squared, so Iโll pull out a negative one. Then weโll have negative one times ๐ฅ to the two plus one power over two plus one, negative ๐ฅ cubed over three. Because weโre looking for the most general form, we canโt forget this constant at the end. Which makes our antiderivative ๐น of ๐ฅ equals ๐ฅ to the eighth power over four minus ๐ฅ to the sixth power over two minus ๐ฅ cubed over three plus ๐.