To integrate \( 7 \tan^2(x) \), we can follow these steps:

1. Start with the original integral:
   \[
   \int 7 \tan^2(x) \, dx
   \]

2. Factor out the constant:
   \[
   7 \int \tan^2(x) \, dx
   \]

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

4. Split the integral:
   \[
   7 \left( \int \sec^2(x) \, dx - \int 1 \, dx \right)
   \]

5. Integrate each term:
   - The integral of \( \sec^2(x) \) is \( \tan(x) \).
   - The integral of 1 is \( x \).

   So, the result is:
   \[
   7 \left( \tan(x) - x \right) + C
   \]
   where \( C \) is the constant of integration.

Thus, the final answer is:
\[
7 \tan(x) - 7x + C
\]

Would you like further details on any step, or do you have any questions?

Here are some related questions for practice:
1. How would you integrate \( \tan^3(x) \)?
2. Can you find the integral of \( \sec^2(x) \tan(x) \)?
3. What is the derivative of \( \tan(x) \)?
4. How would you integrate \( \cot^2(x) \)?
5. Can you solve \( \int \sec^2(x) \, dx \) from 0 to \( \pi/4 \)?

**Tip:** Recognizing common trigonometric identities can simplify complex integrals and make the solution much easier to find!

This guide walks you through the integration of 7tan^2(x), using trigonometric identities and basic integration techniques. Learn how to apply the identity tan^2(x) = sec^2(x) - 1 and find the final solution. Suitable for high school students studying calculus.

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