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The problem asks to calculate \( I_1 \) and \( I_2 \), given the integral expression:

\[
I_n = \int \frac{\cos(nx)}{(\cos x)^n} \, dx
\]

To calculate \( I_1 \) and \( I_2 \), we need to plug in \( n = 1 \) and \( n = 2 \) into the given expression.

### 1. For \( I_1 \):

Substitute \( n = 1 \) into the formula:

\[
I_1 = \int \frac{\cos(x)}{\cos(x)} \, dx = \int 1 \, dx
\]

Thus:

\[
I_1 = x + C
\]

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

### 2. For \( I_2 \):

Substitute \( n = 2 \) into the formula:

\[
I_2 = \int \frac{\cos(2x)}{(\cos x)^2} \, dx
\]

Using the double-angle identity for cosine:

\[
\cos(2x) = 2\cos^2(x) - 1
\]

So the integral becomes:

\[
I_2 = \int \frac{2\cos^2(x) - 1}{\cos^2(x)} \, dx = \int \left(2 - \frac{1}{\cos^2(x)}\right) \, dx
\]

We can split the integral:

\[
I_2 = 2 \int dx - \int \sec^2(x) \, dx
\]

These are both standard integrals:

\[
2 \int dx = 2x
\]
\[
\int \sec^2(x) \, dx = \tan(x)
\]

Thus:

\[
I_2 = 2x - \tan(x) + C
\]

### Final Results:

\[
I_1 = x + C_1
\]
\[
I_2 = 2x - \tan(x) + C_2
\]

Would you like further details or explanations? Here are some follow-up questions for deeper understanding:

1. What trigonometric identities are useful when solving integrals involving cosines?
2. How does the constant of integration play a role in indefinite integrals?
3. Can you find a similar pattern for higher values of \( I_n \)?
4. How can you verify the results of these integrals using differentiation?
5. Why is the secant squared function's integral related to the tangent function?

**Tip:** Always remember to consider trigonometric identities when solving integrals that involve functions like sine, cosine, or secant.

Sea I_n = \int \frac{\cos(nx)}{(\cos x)^n} dx. Calcular I_1 e I_2.

Learn how to calculate I_1 and I_2 for the integral I_n = \int \frac{\cos(nx)}{(\cos x)^n} dx. This problem explores integration techniques and trigonometric identities, suitable for undergraduate mathematics students.

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