<s>[INST] How to integrate $\int\frac1{1+2x}dx$? What approach would be ideal in finding the solution to the integration problem $\int\frac1{1+2x}dx$?
 [/INST] In general
\begin{align}
\int\frac{a}{bx+c}dx=\frac{a}{b}\ln (bx+c)+C
\end{align}
Proof:
Let $u=bx+c$ then $x=\cfrac{u-c}{b}$ and $dx=\cfrac{du}{b}$. Hence
\begin{align}
\int\frac{a}{bx+c}dx=\int\frac{a}{u}\cdot\frac{du}{b}=\frac{a}{b}\int\frac{du}{u}=\frac{a}{b}\ln u+C=\frac{a}{b}\ln (bx+c)+C
\end{align}
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