What is the antiderivative of 1/x²?

The antiderivative, or indefinite integral, of a function is essentially the reverse of taking the derivative. To find the antiderivative of the function
1/x², we can rewrite it using negative exponents:
-1/x^2 = x^{-2}. Now, we will apply the power rule for integration.

The power rule for integration states that for any real number n ≠ -1, the antiderivative of x^n is given by:

∫ x^n dx = (x^(n+1))/(n+1) + C

In our case, since n = -2, we will add 1 to -2, resulting in -1.

Thus, we calculate:

∫ x^{-2} dx = (x^{(-2 + 1)})/(-2 + 1) + C = (x^{-1})/(-1) + C = -1/x + C

Therefore, the antiderivative of 1/x² is:

-1/x + C

where C is the constant of integration. This constant accounts for the fact that antiderivatives are defined up to an arbitrary constant, allowing for an infinite number of functions that differ by a constant to have the same derivative.