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Evaluate the integral \[ \int \frac{\tan(\ln x)\,\tan\!\left(\ln\frac{x}{2}\right)\,\tan(\ln 2)}{x}\,dx. \]

1. $\ln\!\left(\dfrac{\sec(\ln x)}{\sec\!\left(\ln\frac{x}{2}\right)}\right)+C$
2. $\ln(\sec(\ln x))+C$
3. $\ln\!\left(\sec(\ln x)\tan\!\left(\ln\frac{x}{2}\right)\right)+C$
4. $\ln\!\left(\dfrac{\sec(\ln x)}{\sec\!\left(\ln\frac{x}{2}\right)\tan(\ln 2)}\right)+C$

Correct answer: $\ln\!\left(\sec(\ln x)\tan\!\left(\ln\frac{x}{2}\right)\right)+C$

Solution

Let $u=\ln x$, so $du=dx/x$. Then the integral becomes a trigonometric integral in $u$ involving $\tan u$ and $\tan(u-\ln 2)$. Using standard tangent-product identities, it integrates to the logarithm form given in the correct option.

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