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 π not equal to negative one, the integral of π₯ to the power of π with respect to π₯ is equal to one over π plus one times π₯ to the power of π plus one plus an arbitrary constant πΆ. 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 π 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 πΆ.
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 πΆ. This minus πΆ tells us that weβre 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 weβre adding its opposite, and so we have negative one over 10 times π₯ to the power of 10 plus πΆ, 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 πΆ, and that plus πΆ representing an arbitrary constant is very important, but also easy to forget.
