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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.

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  • $\begingroup$ Which textbook is that? $\endgroup$ – J.G. Apr 12 '18 at 8:33
  • $\begingroup$ It is an article. That integration is on page 369. Hopkins, E. (2008). Inequality, happiness and relative concerns: What actually is their relationship? The Journal of Economic Inequality, 6(4), 351-372. $\endgroup$ – shk910 Apr 12 '18 at 9:20
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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.

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