Microeconomics theory and integration by parts (proof).

Question

I understand this part: $\int x^\prime (z)F(z) dz+\int x(z)f(z)dz=\int zf(z)dz \rightarrow \int \frac {dx}{dz} F(z) dz+\int x(z)\frac {dF(z)}{dz} dz=\int zf(z)dz \rightarrow x(z)F(z)= \int_{0}^z tf(t)dt$

Then, the author says obtain the following solution by integration by parts. $x(z)=z-\frac {\int_0^zF(t)dt}{F(z)}$

I don't know which term should I integrate so that I get to that result.

Thank you in advance.

Answer 1

Integrate the $f(t)$ (a primitive of which is $F(t)$) and differentiate the $t$. This yields

\begin{align*} x(z) F(z) & = \int_{0}^{z}{t f(t)dt} \\ & = \big[ t F(t) \big ]_{t=0}^{t=z}-\int_{0}^{z}{F(t)dt} \\ & = z F(z)-\int_{0}^{z}{F(t)dt}. \end{align*}

Dividing by $F(z)$ on both sides yields the result.