Video Transcript
Determine the antiderivative of
negative 𝑥 to the power of nine with respect to 𝑥. This is also called the indefinite
integral of negative 𝑥 to the power of nine.
For 𝑛 not equal to negative one,
the integral of 𝑥 to the power of 𝑛 with respect to 𝑥 is equal to one over 𝑛
plus one times 𝑥 to the power of 𝑛 plus one plus an arbitrary constant 𝐶. 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 𝑘 equal to minus one,
we can see that the antiderivative of negative 𝑥 to the power of nine with respect
to 𝑥 is just the negative of the antiderivative of 𝑥 to the power of nine with
respect to 𝑥.
And taking 𝑛 equal to nine, we can
see that the antiderivative of 𝑥 to the power of nine using our formula is one over
nine plus one times 𝑥 to the power of nine plus one plus 𝐶.
And so putting that together with
the minus sign that we have, we get negative one over 10𝑥 to the power of 10 minus
𝐶. This minus 𝐶 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 𝑥 to the power of 10 plus 𝐶, which is more
conventional.
Our final answer is that the
antiderivative of negative 𝑥 to the power of nine with respect to 𝑥 is negative
one over 10𝑥 to the power of 10 plus 𝐶, and that plus 𝐶 representing an
arbitrary constant is very important, but also easy to forget.
