1. The problem statement, all variables and given/known data Here's the problem with the solution provided: 2. Relevant equations Fundamental Theorem of Calculus (FToC) 3. The attempt at a solution So I understand everything up to where I need to take the derivative of the integral(s). Couple of things I know is that the derivative of the CDF, F(T) is the PDF, f(t). So naturally the integral of the PDF would be the CDF, right? So for the first integral, ct∫f(x)dx [0,t] I would need to use the product rule for this I think, so I'd have: c∫f(x)dx [0,t] + ctf(t) And since the integral of the PDF is the CDF, this would be: cF(t) + ctf(t) right? That's what they have so far For the second integral, -c∫xf(x)dx[0,t] The derivative of this would just be -ctf(t) by the Fundamental Theorem of Calculus. The third integral, k∫xf(x)dx [t,∞] I believe I would need to switch the limits of integration for this to be able to differentiate using the FToC: -k∫xf(x)dx [∞,t] Then, differentiating this: -ktf(t) For the fourth integral I'm not sure what they did. I thought we just needed to switch the limits of integration again to get kt∫f(x)dx [∞,t] And then differentiate this to get k∫f(x)dx [∞,t] + ktf(t) = kF(t) + ktf(t) so in total, I'd have: cF(t) + ctf(t) - ctf(t) - ktf(t) + kF(t) + ktf(t) which matches what they have except for the last two terms. I'm not sure what's going on there, can someone explain?