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

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

<p>Dividing by $F(z)$ on both sides yields the result.</p>

CC BY-SA 3.0
2018-04-12T09:18:39.160Z
2018-04-12T09:18:39.160Z
Oliv
<p><a href="https://i.sstatic.net/qSKrb.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/qSKrb.png" alt="enter image description here"></a></p>

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

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

<p>I don't know which term should I integrate so that I get to that result.  </p>

<p>Thank you in advance. </p>

CC BY-SA 3.0
2018-04-12T08:33:32.150Z
Which textbook is that?
J.G.
CC BY-SA 3.0
2018-04-12T09:20:23.900Z
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.
shk910
<microeconomics>
Microeconomics theory and integration by parts (proof).
shk910
# Microeconomics theory and integration by parts (proof).

(Asked by: username_0 on 2018-04-12)

[![enter image description here](https://i.sstatic.net/qSKrb.png)](https://i.sstatic.net/qSKrb.png)

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.

**Tags:** microeconomics
> **Comments:**
> **username_1**: Which textbook is that?  
> **username_0**: 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.  

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## 1 answers

### Answer by username_2 on 2018-04-12. Score: 4 | Accepted answer

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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