Video: The Power Rule of Integration

Determine ∫ βˆ’π‘₯⁹ dπ‘₯.

02:03

Video Transcript

Determine the antiderivative of negative π‘₯ to the power of nine with respect to π‘₯. This is also called the indefinite integral of negative π‘₯ to the power of nine.

For 𝑛 not equal to negative one, the integral of π‘₯ to the power of 𝑛 with respect to π‘₯ is equal to one over 𝑛 plus one times π‘₯ to the power of 𝑛 plus one plus an arbitrary constant 𝐢. This is very nearly the form of what we have, but we have this negative sign here.

We can use the fact that the antiderivative of a scalar multiple of a function is just that scalar multiple of the antiderivative of the function. And so, with π‘˜ equal to minus one, we can see that the antiderivative of negative π‘₯ to the power of nine with respect to π‘₯ is just the negative of the antiderivative of π‘₯ to the power of nine with respect to π‘₯.

And taking 𝑛 equal to nine, we can see that the antiderivative of π‘₯ to the power of nine using our formula is one over nine plus one times π‘₯ to the power of nine plus one plus 𝐢.

And so putting that together with the minus sign that we have, we get negative one over 10 π‘₯ to the power of 10 minus 𝐢. This minus 𝐢 tells us that we’re subtracting an arbitrary constant from our function, and this constant could be positive or negative, so we might as well say that we are adding its opposite.

So rather than subtracting this arbitrary constant, we can say that we’re adding its opposite, and so we have negative one over 10 times π‘₯ to the power of 10 plus 𝐢, which is more conventional.

Our final answer is that the antiderivative of negative π‘₯ to the power of nine with respect to π‘₯ is negative one over 10 π‘₯ to the power of 10 plus 𝐢, and that plus 𝐢 representing an arbitrary constant is very important, but also easy to forget.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.