What is the idea to integrate this equation from G&R?

Question

$$\int {\mathrm dx \over {x z^m}}= {1 \over {z^{m-1}a(m-1)}} + {1 \over {a}} \int {\mathrm dx \over xz^{m-1}},$$ where $z = a+bx$.

I tried integration by parts method but cannot get the powers right! What is the idea to get achieve them?

Answer 1

Well, you usually don't do that step naturally (and that is a reason to keep such entry in a table). But here is why it is true. First, simplify the integrand in the left-hand side minus the integrand in the right-hand side: $$ \frac{1}{xz^m}-\frac{1}{axz^{m-1}}=-\frac{b}{az^m}. $$ Then just integrate, $$ \int -\frac{b}{a(a+b x)^m}\,dx=\frac{1}{a(m-1)(a+bx)^{m-1}}+C. $$ The constant $C$ can be omitted if one rearranges as in the table entry.