Question Video: Finding the Integration of a Function Using the Power Rule for Integration with a Negative Exponent Mathematics • Higher Education

Determine ∫ −(2/7)𝑥⁻⁹ d𝑥.

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Video Transcript

Determine the indefinite integral of negative two over seven multiplied by 𝑥 to the power of negative nine with respect to 𝑥.

In our integral, we simply have a power function. So we can use the power rule for integration in order to find this integral. It tells us that the indefinite integral of 𝑥 to the power of 𝑛 with respect to 𝑥 is equal to 𝑥 to the power of 𝑛 plus one over 𝑛 plus one plus 𝐶. In our case, we’re integrating negative two-sevenths 𝑥 to the power of negative nine with respect to 𝑥. So our power of 𝑥 is negative nine. We can start by writing down our constant which is negative two-sevenths.

Then since our value of 𝑛 is negative nine, we next need to write 𝑥 to the power of 𝑛 plus one over 𝑛 plus one. So that’s 𝑥 to the power of negative nine plus one which is 𝑥 to the power of negative eight over negative nine plus one. So that’s negative eight. Then we mustn’t forget to add our constant of integration 𝐶. For our final step here, we just need to simplify. And we obtain our solution which is that the indefinite integral of negative two-seventh multiplied by 𝑥 to the power of negative nine with respect to 𝑥 is equal to 𝑥 to the power of negative eight over 28 plus 𝐶.

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