To solve the integral:

$\int \frac{3 \tan^2(x)}{\sec(x)} \, dx$

we first express the integrand in simpler terms using trigonometric identities.

Step 1: Simplify the expression using trigonometric identities

We know that:

$\sec(x) = \frac{1}{\cos(x)} \quad \text{and} \quad \tan(x) = \frac{\sin(x)}{\cos(x)}$

Thus, we can rewrite $\tan^2(x)$ as:

$\tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}$

Now substitute these expressions into the original integral:

$\int \frac{3 \tan^2(x)}{\sec(x)} \, dx = \int \frac{3 \frac{\sin^2(x)}{\cos^2(x)}}{\frac{1}{\cos(x)}} \, dx$

Simplifying the fraction:

$= \int \frac{3 \sin^2(x)}{\cos(x)} \, dx$

Step 2: Use a substitution

Let’s try a substitution. Let $u = \cos(x)$, then:

$\frac{du}{dx} = -\sin(x) \quad \text{or} \quad dx = \frac{du}{-\sin(x)}$

This changes the integral into:

$-3 \int \frac{(1 - u^2)}{u} \, du$

Step 3: Simplify the new integral

Now break it into two integrals:

$-3 \int \frac{1}{u} \, du + 3 \int u \, du$

Step 4: Integrate each part

The integrals are straightforward:

$-3 \ln |u| + \frac{3u^2}{2} + C$

Step 5: Substitute $u = \cos(x)$ back into the result

Now, replace $u$ with $\cos(x)$:

$-3 \ln |\cos(x)| + \frac{3}{2} \cos^2(x) + C$

Thus, the final solution is:

$-3 \ln |\cos(x)| + \frac{3}{2} \cos^2(x) + C$

Would you like further details or clarification on any step?

Here are 5 related questions you might find interesting:

1. How do you compute integrals involving $\sec(x)$ and $\tan(x)$ generally?
2. What are common trigonometric substitutions used for integrals?
3. How does one handle integrals with powers of trigonometric functions?
4. Why do we use absolute values in logarithmic results?
5. Can integration by parts be used in trigonometric integrals?

Tip: Substitution is a powerful tool for integrals involving complex trigonometric expressions!