antiderivative (1/x^2)?

Question

how do i do the antiderivative when x is in the denominator?

(1/x^2)

Answer 1

Bring it to the top
1/x^2 = x^-2
The antiderivative is
x^(-2+1)/(-2+1) + C
= x^(-1)/(-1) + C
= -1/x + C

Answer 2

First, you have to convert the problem such that there is no denominator; you want to move x^2 up to the numerator, and when doing so, it is no longer a fraction.

1/x^2 = x^(-2).

Now, you take the antiderivative as normal for powers, using the reverse power rule for derivatives. Reminder that if

f(x) = x^n and F(x) is the antiderivative of f(x), then
F(x) = [x^(n+1)]/(n+1)

We do the same thing for this case. If

f(x) = x^(-2) then
F(x) = [x^(-1)]/(-1) + C = -x^(-1) + C = -1/(x^1) + C = -1/x + C

(We ultimately want no negative powers even though we had to utilize them to solve this problem; our final answer should preferably be in all positive powers, hence why I did that).

Another side note is that when taking the antiderivative, you're getting the general antiderivative, so we have to add a constant C.