Originally Posted by

**Soroban** Hello, ryu1!

If you know the correct answer, you can *see* what you are missing, can't you?

Okay, back to basics.

Formula: .$\displaystyle \int\frac{du}{u^2+b^2} \;=\;\frac{1}{b}\arctan\left(\frac{u}{b}\right) + C$

We have: .$\displaystyle \int\frac{dx}{(ax)^2 + b^2}$

That is: .$\displaystyle u \,=\,ax \quad\Rightarrow\quad du \,=\,a\,dx \quad\Rightarrow\quad dx \,=\,\tfrac{1}{a}du$

Substitute: .$\displaystyle \int\frac{\frac{1}{a}du}{u^2+b^2} \;=\;\frac{1}{a}\int\frac{du}{u^2+b^2} $

. . . . . . . . $\displaystyle =\;\frac{1}{a}\cdot\frac{1}{b}\arctan\left(\frac{u }{b}\right)+C \;=\;\frac{1}{ab}\arctan\left(\frac{u}{b}\right)+C$

Back-substitute: .$\displaystyle \frac{1}{ab}\arctan\left(\frac{ax}{b}\right)+C$