1. **Problem statement:** Find the constant $c$ given the probability density function (pdf) $f(x) = c x e^{-2x}, x > 0$. The total probability must integrate to 1: $$\int_0^{\infty} c x e^{-2x} dx = 1$$ 2. Calculate the integral without $c$: $$\int_0^{\infty} x e^{-2x} dx$$ 3. Use integration or gamma function formula: $$\int_0^{\infty} x e^{-2x} dx = \frac{1}{2^2} = \frac{1}{4}$$ 4. Substitute into the probability constraint: $$c \times \frac{1}{4} = 1 \implies c = 4$$ 5. **Final answer:** $$\boxed{c = 4}$$