Help with integration

[Marinagp13]

Hello!
I would like you to help me integrate this function:

My unknowns are W and x, and I'd like to plot x vs. W. The rest are constants.
Can anyone give me an expression like 'W=....' without any integrals? I'd really appreciate it.
Thanks in advance.

 

[HallsofIvy]

If "all the rest are constants" let's write this as $\displaystyle \frac{dx_a}{dW}= \frac{A(1- x_a)}{1+ B(1- x_a)}$. Then an obvious simplification is to let $\displaystyle y= 1- x_a$ so that $\displaystyle x_a= 1-y$, $\displaystyle \frac{dx_a}{dy}= -1$, and $\displaystyle \frac{dx_a}{dW}= \frac{dy}{dW}\frac{dx_a}{dy}= -\frac{dy}{dW}= \frac{Ay}{1+ By}$.
That is easily separable as $\displaystyle \frac{(1+ By)dy}{Ay}= \frac{1}{A}\frac{dy}{y}+ \frac{B}{A}dy= -dW$. Integrating, $\displaystyle \frac{1}{A} ln(|y|)+ \frac{B}{A}y= -W+ constant$.
$\displaystyle W= -\frac{1}{A}ln(|y|)- \frac{B}{A}y+ constant$.

And since $\displaystyle y= 1- x_a$,
$\displaystyle W= -\frac{1}{A}ln(|1- x_a|)- \frac{B}{A}(1- x_a)+ constant$.

My "A" is your "$\displaystyle \frac{k_wC_{A,0}}{F_{A,0}}$" and my "B" is your "$\displaystyle KC_{A,0}$".

 

[topsquark]

Hello!
I would like you to help me integrate this function:
View attachment 12606
My unknowns are W and x, and I'd like to plot x vs. W. The rest are constants.
Can anyone give me an expression like 'W=....' without any integrals? I'd really appreciate it.
Thanks in advance.

It would seem to me that $\displaystyle k_w$ would also change with changing W? We need to know what $\displaystyle k_w$ would be and how it would change.

-Dan