This is an integral of $\int \tan\left(\frac{x}{2}\right) dx$. Let's solve it step by step.

Solution:

1. Use the substitution:
   Let $u = \frac{x}{2}$.
   Then, $du = \frac{1}{2}dx$, or equivalently $dx = 2du$.

   The integral becomes: $\int \tan\left(\frac{x}{2}\right) dx = \int \tan(u) \cdot 2 du = 2 \int \tan(u) du$

2. Integral of $\tan(u)$:
   Recall that: $\int \tan(u) du = -\ln|\cos(u)| + C$

   So: $2 \int \tan(u) du = -2 \ln|\cos(u)| + C$

3. Back-substitute $u = \frac{x}{2}$:
   Replace $u$ with $\frac{x}{2}$: $-2 \ln|\cos(u)| + C = -2 \ln\left|\cos\left(\frac{x}{2}\right)\right| + C$

Final Answer:

$\int \tan\left(\frac{x}{2}\right) dx = -2 \ln\left|\cos\left(\frac{x}{2}\right)\right| + C$

Would you like a deeper explanation or clarification of any step?

Related Questions:

1. How do we derive the integral of $\tan(u)$?
2. What are other substitution techniques for trigonometric integrals?
3. How can we verify the result by differentiation?
4. What happens if we write $\tan(u)$ in terms of $\sin(u)$ and $\cos(u)$?
5. Can this integral be evaluated numerically for a specific value of $x$?

Tip:

When solving integrals involving trigonometric functions, substitution is often a powerful tool to simplify the expression!