$$ {{\displaylines\int_{}^{}\!\tan^2\,x\text{ dx}\\ }} $$

To find the integral of $\tan^2 x$, we can use the trigonometric identity:
\[ \tan^2 x = \sec^2 x - 1 \]

So, the integral becomes:
\[ \int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx \]

We can split this into two separate integrals:
\[ \int (\sec^2 x - 1) \, dx = \int \sec^2 x \, dx - \int 1 \, dx \]

Now, we integrate each term separately:
\[ \int \sec^2 x \, dx = \tan x + C_1 \]
\[ \int 1 \, dx = x + C_2 \]

Combining these results, we get:
\[ \int (\sec^2 x - 1) \, dx = \tan x - x + C \]

where $C$ is the constant of integration, which is $C = C_1 - C_2$.

Thus, the final answer is:
\[ \int \tan^2 x \, dx = \tan x - x + C \]

Would you like more details or have any questions about this solution?

Here are 5 related questions you can ask:
1. How do you integrate $\sec^2 x$?
2. What is the integral of $\tan x$?
3. Can you explain how to use substitution in integrals?
4. How do you find the derivative of $\tan x$?
5. What are some applications of integrals in real life?

**Tip:** When dealing with trigonometric integrals, using identities to simplify the integrand can often make the integration process easier.

Discover how to integrate tan^2(x) step-by-step using trigonometric identities and integration techniques. This detailed solution covers the integral of tan^2(x), providing insights into calculus and trigonometry.

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