The antiderivative of 1/x² is -1/x + C, where C is the constant of integration.
To understand why, let’s recall the basic rule of antiderivatives. The antiderivative, or indefinite integral, of a function f(x) gives us a new function whose derivative is f(x).
In this case, we can rewrite 1/x² as x-2. Using the power rule for antiderivatives, which states that the antiderivative of xn is (xn+1)/(n+1) plus a constant C, we compute:
- Setting n = -2, we get:
- (x-2+1)/( -2 + 1) = (x-1)/(-1) = -1/x
Thus, we add the constant of integration C to encompass all possible antiderivatives. So, the final result is:
∫(1/x²) dx = -1/x + C