Find $\displaystyle\int \frac{\tan(\ln x)\,\sec^{2}(\ln x)}{x}\,dx$.

The key is recognizing that $\ln x$ is "hiding" inside the trig functions—a classic setup for $u$-substitution. Notice that the derivative of $\ln x$ is $\frac{1}{x}$, which appears in the denominator, and $\sec^2(\ln x)$ is the derivative of $\tan(\ln x)$. This means you have a composition where the inner function's derivative is already present, letting you collapse the integral into a simple power form.

The key is recognizing that $\ln x$ is "hiding" inside the trig functions—a classic setup for $u$-substitution. Notice that the derivative of $\ln x$ is $\frac{1}{x}$, which appears in the denominator, and $\sec^2(\ln x)$ is the derivative of $\tan(\ln x)$. This means you have a composition where the inner function's derivative is already present, letting you collapse the integral into a simple power form.