First order, parallel reaction integration
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.