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The problem involves two parts of definite and indefinite integrals:

### Part i) \( \int \tan{x} \, dx \)

To solve this integral, we can rewrite it using a known formula:
\[
\int \tan{x} \, dx = \ln|\sec{x}| + C
\]
where \(C\) is the constant of integration. This result comes from expressing \(\tan{x} = \frac{\sin{x}}{\cos{x}}\) and using substitution.

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### Part ii) \( \int_1^2 (3x)^2 \, dx \)

We need to evaluate the definite integral:
\[
\int_1^2 (3x)^2 \, dx
\]
First, simplify the integrand:
\[
(3x)^2 = 9x^2
\]
So the integral becomes:
\[
\int_1^2 9x^2 \, dx
\]
Use the power rule for integration:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\]
Applying this to \(9x^2\):
\[
\int 9x^2 \, dx = 9 \cdot \frac{x^3}{3} = 3x^3
\]

Now, apply the limits from 1 to 2:
\[
\left[ 3x^3 \right]_1^2 = 3(2^3) - 3(1^3) = 3(8) - 3(1) = 24 - 3 = 21
\]

Thus, the value of the definite integral is:
\[
\boxed{21}
\]

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Let me know if any step needs clarification! Here are 5 related questions:

1. How do you derive the formula for the integral of \( \tan{x} \)?
2. Can you generalize the power rule for integrals to fractional exponents?
3. How does changing the limits of integration affect the value of a definite integral?
4. What other trigonometric integrals are commonly encountered?
5. How do substitution and integration by parts help solve more complex integrals?

**Tip:** Always double-check definite integrals by ensuring you apply the limits correctly!

Evaluate the integral: i) ∫ tan(x) dx ii) ∫ from 1 to 2 of (3x)^2 dx

Learn how to solve both indefinite and definite integrals involving the tangent function and polynomials. This problem includes using logarithmic integration for tan(x) and applying the power rule for (3x)^2, suitable for college-level calculus students.

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