Quick question on integration by parts

Question

Suppose I want to find the indefinite integral for $\frac {a(x)}{b(x)}$ where $a(x)=p(x)q(x)$, am I correct in solving it this way: $$\begin{align} \int \frac {a(x)}{b(x)} dx&=\frac1{b(x)}\int a(x) \ dx-\int\biggl(\frac1{b(x)}\biggr)'\biggl(\int a(x) \ dx\biggr) \ dx\\ &=\frac1{b(x)}\int p(x)q(x) \ dx-\int\biggl(\frac1{b(x)}\biggr)'\biggl(\int p(x)q(x) \ dx\biggr) \ dx\\ &=\frac1{b(x)}\biggl(p(x)\int q \ dx-\int p'(x)\Bigl(\int q(x) \ dx\Bigr) \ dx\biggr)-\int\biggl(\frac1{b(x)}\biggr)'\biggl(p(x)\int q(x) \ dx-\int p'(x)\Bigl(\int q(x) \ dx\Bigr) \ dx\biggr) \ dx \end{align} $$ Is this integral correct? Or are there principles I am overlooking? And also, are there other things that I should keep in mind?