My Math Forum Integration, confused

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 August 2nd, 2016, 09:16 AM #1 Newbie   Joined: Aug 2016 From: uk Posts: 1 Thanks: 0 Integration, confused I am reading a research paper and I could not understand how the author derived the final equation using integration. Equation A: $Y_L=\frac{1}{1-\beta}\left [\displaystyle\int_0^{N_L}x_L(j)^{1-\beta}\,dj\right ]L^{\beta }$ Equation B: $x_L=\left [\frac{p_L}{\chi_L(j)}\right ]^{1/\beta}L$ Equation C: $Y_L=\frac{1}{1-\beta}p_L^{\frac{1-\beta}{\beta}}N_LL$ where, - $N_L$ is the number of varieties of machines - $x_L$ is the range of machines, so $x_L(j)$ is a machine type $(j)$ - $\chi_L(j)$ is the price of machine type $(j)$ The author uses the equation B in equation A and derives the equation C after integration. It appears, the author replaces $x_L(j)$ in equation A with the equation for $x_L$ which is the equation B. Can anyone please help understand how this could be done?
 August 2nd, 2016, 03:50 PM #2 Global Moderator   Joined: May 2007 Posts: 6,627 Thanks: 622 Statement A is puzzling. Is j the variable of integration? The description has j a discrete variable.

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