Solve the integral equation for $f$.

Question

Find all functions $f$ that satisfy: $$\int f(x)dx \cdot \int (1/f(x))dx = -1$$ So far, I have tried a handful of methods. I have substituted a variable $u$ for $f(x)$, I have substituted an antiderivative $F(x)$ and $G(x)$ for the first and second integrands, respectively. Using this second method, I thought I was on the right track until coming to: $$f(x) = \pm 1/\int f(x)dx$$ Any helpful replies are appreciated, but this problem is solvable using relatively basic methods. This problem is from Stewart's Early Transcendentals.