Find the integral involving secant and tangent. (Remember to use absolute values where appropriate.)

 *tan is in parens

fraction: 3 (tan(*x*))2

sec(*x*)

 *dx*
 

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!

Find the integral involving secant and tangent. 

(3 * tan(x)^2) / sec(x) dx

Learn how to solve the integral (3 * tan^2(x)) / sec(x) dx using trigonometric identities and substitution. This detailed solution includes step-by-step explanations on simplifying the expression, applying the substitution method, and handling logarithmic integration results. Suitable for students at an advanced high school or undergraduate calculus level.

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