I thought that $$ \int_{-\infty}^C t f_X(t) dt = E[X\vert X\leq C] $$ because the LHS is the expectation but restricted to the domain where $X\leq C$
But in an Economics textbook (Mas-Collel Whinston and Greene page 447) I come across an equation that seems to imply $$ \int_{-\infty}^C t f_X(t) dt = F_X(C)\cdot E[X\vert X\leq C] $$ Which means I must be wrong about one of the equations above?
So am I wrong about the first equation (what I think the conditional expectation is as an integral), or am I wrong about the second equation?
if the second equation is true, can someone help me show it (or point me to a source)?
Note Assume $f_X(x)>0$ for all $x$ in the domain of $X$.
I suspect that the issue is that you're confusing $\mathbb{E}[X\mid X\leq C]$ and $\mathbb{E}[X1_{X\leq C}]$. These two quantities are closely related but are not the same in general.
Recall that $$ \mathbb{E}[X\mid X\leq C]=\frac{\mathbb{E}[X1_{X\leq C}]}{\mathbb{P}(X\leq C)}$$ provided that $\mathbb{P}(X\leq C)>0$. If $X$ has CDF $F_X$ and pdf $f_X$, then $$ \mathbb{E}[X1_{X\leq C}]=\int_{-\infty}^Ctf_X(t)\;dt $$ and $$ \mathbb{P}(X\leq C)=F_X(C)$$ Therefore $$\mathbb{E}[X\mid X\leq C]=\frac{\int_{-\infty}^Ctf_X(t)\;dt}{F_X(C)}$$ or $$ \int_{-\infty}^Ctf_X(t)\;dt=F_X(C)\mathbb{E}[X\mid X\leq C]$$ as your book states.