The table of integrals says that
\begin{equation*} \int \frac{dx}{a^{2}+x^{2}}=\frac{1}{a}\arctan\frac{x}{a}+C \end{equation*}
where $C$ is a constant. What's wrong with my proof?
$$ \begin{align*} y &= \arctan\frac{x}{a} \\\ a\tan y&=x \\\ \tan y &= \frac{x}{a} \\\ \frac{d}{dx} \tan y &= \frac{d}{dx} \frac{x}{a} \\\ \sec^{2}y \frac{dy}{dx} &= \frac{1}{a} \\\ \frac{dy}{dx} &= \frac{1}{a} \frac{1}{\sec^{2}y} \\\ &= \frac{1}{a} \frac{1}{1 + \tan^{2} y} \\\ \frac{dy}{dx}&= \frac{1}{a} \frac{1}{1 + (\frac{x}{a})^{2}} \\\ \int \frac{dy}{dx}\\,dx &= \int \frac{1}{a} \frac{1}{1 + (\frac{x}{a})^{2}}\\,dx\\\ \arctan\frac{x}{a} &= \frac{1}{a} \int \frac{1}{1 + (\frac{x}{a})^{2}}\\,dx \\\ \end{align*} $$
Is it right so far? Any help appreciated!