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The integral to evaluate is:

\[
\int \tan^8(x) \cos^9(x) \, dx
\]

### Approach:
We can use a substitution method to simplify this integral. Specifically, we know the identity:

\[
\tan(x) = \frac{\sin(x)}{\cos(x)}
\]

Thus, rewriting \(\tan^8(x)\) as \(\left(\frac{\sin(x)}{\cos(x)}\right)^8\) and simplifying the trigonometric expressions could be useful. Let's proceed step by step.

1. Use the substitution \( u = \sin(x) \), so that \( du = \cos(x) \, dx \).

### Step-by-step process:
We will compute the integral based on these steps. Let me do that for you.The evaluated integral is:

\[
\int \tan^8(x) \cos^9(x) \, dx = \frac{\sin^9(x)}{9} + C
\]

where \( C \) is the constant of integration.

Would you like further details or clarifications on this? Here are some related questions to explore:

1. How does the substitution \( u = \sin(x) \) simplify the integral process?
2. What other trigonometric identities can be useful for integrating powers of tangent and cosine?
3. How would the result change if the powers of \(\tan(x)\) and \(\cos(x)\) were different?
4. Could this integral be solved using a different method, such as integration by parts?
5. How do integrals of trigonometric functions differ when the powers are odd versus even?

**Tip:** When dealing with integrals involving both \(\tan(x)\) and \(\cos(x)\), substitutions involving \(\sin(x)\) and its derivatives often simplify the process significantly.

Evaluate the integral: ∫ tan^8(x) cos^9(x) dx

Learn how to evaluate the integral of tan^8(x) cos^9(x) using trigonometric identities and substitution methods. This problem involves key calculus concepts suitable for undergraduate and advanced high school students.

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