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