First order, parallel reaction integration I have parallel equations: 
\begin{align}\ce{A &amp;-&gt;[k_b] B},&amp;\ce{A &amp;-&gt;[k_c] C}.\end{align}
I understand how to determine formation of $\ce{A}$ (with the integration) but I really keep going in circles with integrating $\ce{B}$. If I can have some assistance in how to integrate $\ce{B}$, then I will be fine for integrating $\ce{C}$.

This is what I have:
\begin{align}
\frac{\mathrm{d}[\ce{B}]}{\mathrm{d}t} 
&amp;= k_b[\ce{A_0}] \cdot \mathrm{e}^{(-k_b+k_c)t}\\
\int_{[\ce{A_0}]}^{[\ce{A}]}\frac{\mathrm{d}[B]}{\mathrm{d}t} 
&amp;= \int k_b\cdot[A] \mathrm{d}t\\
\ln\frac{[\ce{B}]}{[\ce{A_0}]} 
&amp;= \int k_b\cdot[\ce{A_0}] \cdot \mathrm{e}^{-(k_b+k_c)\cdot t}\mathrm{d}t\\
\end{align}

To save the embarrassment, once I raise to the "$\mathrm{e}$" of both sides, I get 
\begin{align}
\frac{[\ce{B}]}{[\ce{A_0}]} 
&amp;= \mathrm{e}^{\frac{k_b\cdot[\ce{A_0}]}{(k_b+k_c)}t} \cdot \mathrm{e}^{\mathrm{e}^ {-(k_b+k_c)t}}\\
[\ce{B}] &amp;=[\ce{A_0}] \mathrm{e}^{\frac{k_b\cdot[\ce{A_0}]}{k_b+k_c}t} \cdot \mathrm{e}^{\mathrm{e}^{-(k_b+k_c)t}}\\
\end{align}

Can someone show me the steps of integrating the $\int k_b\cdot[\ce{A}] \mathrm{d}t$?
Very basic steps are available in solutions, but it still does not fully show the integration/algebra and I am just very lost.
Start by solving the rate equation for $\ce{[A]}$:
\begin{align}
\frac{\mathrm{d}\ce{[A]}}{\mathrm{d}t} 
&amp;= -k_\mathrm{B}\ce{[A]} - k_\mathrm{C}\ce{[A]}&amp;
\Longleftrightarrow&amp;&amp;
\ce{[A]} 
&amp;= \ce{[A]_0}\exp\left(-(k_\mathrm{B}+k_\mathrm{C}\right)t).
\end{align}
We can then solve the rate equation for $\ce{[B]}$:
\begin{align}
\frac{\mathrm{d}\ce{[B]}}{\mathrm{d}t} &amp;= k_\mathrm{B}\ce{[A]} \\
&amp;= k_\mathrm{B}\ce{[A]_0}\exp\left(-(k_\mathrm{B} + k_\mathrm{C}\right)t)\\
\ce{[B]}-\ce{[B]_0} 
&amp;= \int_{\ce{[B]_0}}^{\ce{[B]}}\mathrm{d}\ce{[B]}' \\
&amp;= k_\mathrm{B}\ce{[A]_0}\int_0^t\mathrm{d}t'\,\exp\left(-(k_\mathrm{B}+k_\mathrm{C}\right)t')\\
 &amp;= \frac{k_\mathrm{B}\ce{[A]_0}}{k_\mathrm{B} + k_\mathrm{C}}[1-\exp\left(-(k_\mathrm{B}+k_\mathrm{C}\right)t)].
\end{align}
I use primes to distinguish the variable of integration from the limits of integration.