# Stuck on tough integral

I am trying to solve what looked like a simple integral but I got a bit stuck. The integral is : $$\int_0^x \frac{ab(1-e^{-ct})}{d-\frac{b(1-e^{-ct})}{c}}dt$$

I tried change the change of variable $y=1-e^{-ct}$ and got :

$$\int_0^{1-e^{-ct}} \frac{1}{y+\frac{cd}{by}-\frac{cd}{b}-1}dt$$

Now I am looking into changing again the variable $z=y+\frac{cd}{by}-\frac{cd}{b}-1$ but I am troubled by the lower limit of integration (when y=0). Can someone give this a try maybe you spot a better way of solving the thing. Thanks.

• With your first change of variable I got $ab\int\limits_{0}^{1-{{e}^{-cx}}}{\frac{ydy}{(cd-by)(1-y)}}$. Now do partial fractions and integrate. – Paul Sep 24 '14 at 15:14
• yes, I made a stupid mistake. Thanks Paul. – KAT Sep 24 '14 at 17:30

$$ac \int \dfrac{y}{\frac{cd}{b}-y} \dfrac{1}{(1-y)}dy$$
$$\int \dfrac{y}{(A-y)(B-y)} dy = \dfrac{1}{A-B}\int \dfrac{B}{B-y} - \dfrac{A}{A-y} dy$$