$$ \int\frac{5\sec^2\theta}{\tan^2\theta} $$$$ d\theta $$

To solve the integral:

\[
\int \frac{5 \sec^2 \theta}{\tan^2 \theta} \, d\theta
\]

we can start by expressing everything in terms of $\tan \theta$, as that will simplify the integrand.

### Step 1: Rewrite using $\tan \theta$
We know that:
\[
\frac{d}{d\theta} (\tan \theta) = \sec^2 \theta
\]
Thus, we can substitute $ u = \tan \theta $, so that $ du = \sec^2 \theta \, d\theta $.

Now the integral becomes:
\[
5 \int \frac{1}{u^2} \, du
\]

### Step 2: Solve the integral
The integral of $ \frac{1}{u^2} $ is a standard integral:
\[
\int \frac{1}{u^2} \, du = -\frac{1}{u}
\]

Thus, the integral becomes:
\[
5 \left( -\frac{1}{u} \right) = -\frac{5}{u}
\]

### Step 3: Substitute $ u = \tan \theta $
Substituting back $ u = \tan \theta $, we get:
\[
-\frac{5}{\tan \theta} + C
\]

Finally, using the identity $ \frac{1}{\tan \theta} = \cot \theta $, we can rewrite the solution as:
\[
-5 \cot \theta + C
\]

### Final Answer:
\[
\int \frac{5 \sec^2 \theta}{\tan^2 \theta} \, d\theta = -5 \cot \theta + C
\]

---

Do you need further clarification or details on any step?

Here are some related questions for further exploration:
1. What is the integral of $ \int \sec^2 \theta \, d\theta $?
2. How does the substitution method work in trigonometric integrals?
3. What is the derivative of $ \cot \theta $?
4. How can we express $ \tan \theta $ in terms of sine and cosine?
5. What are the common integrals involving $ \sec \theta $ and $ \tan \theta $?

**Tip:** Always check if a trigonometric identity can simplify the integrand before attempting substitution.

$$ \int\frac{5\sec^2\theta}{\tan^2\theta} \, d\theta $$

Learn how to solve the integral $$\int\frac{5\sec^2\theta}{\tan^2\theta}\,d\theta$$ by applying substitution and trigonometric identities. This step-by-step solution demonstrates core trigonometry concepts and integration techniques. Suitable for high school students in Grades 11-12.

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