https://file.aiquickdraw.com/mathbot/file/1726933844060l9dfna8x.jpg

Let's carefully go through the steps shown in the image to check the correctness of the solution.

We are given the integral:

\[
\int \frac{\cos(2x) + \sin^2(x)}{\cos^2(x)} \, dx
\]

### Step 1:
The expression is split into two terms:

\[
\int \frac{\cos(2x)}{\cos^2(x)} \, dx + 2 \int \frac{\sin^2(x)}{\cos^2(x)} \, dx
\]

This is correct since we can split the integral based on the sum of terms.

### Step 2:
The term \(\frac{\sin^2(x)}{\cos^2(x)}\) is recognized as \(\tan^2(x)\), so:

\[
\int \frac{\cos(2x)}{\cos^2(x)} \, dx + 2 \int \tan^2(x) \, dx
\]

This is a correct transformation.

### Step 3:
Using the identity \(\cos(2x) = \cos^2(x) - \sin^2(x)\), the first integral becomes:

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

Since \(\frac{\sin^2(x)}{\cos^2(x)} = \tan^2(x)\), this simplifies to:

\[
\int (1 - \tan^2(x)) \, dx + 2 \int \tan^2(x) \, dx
\]

### Step 4:
Next, the integral becomes:

\[
\int 1 \, dx - \int \tan^2(x) \, dx + 2 \int \tan^2(x) \, dx
\]

This simplifies to:

\[
\int 1 \, dx + \int \tan^2(x) \, dx
\]

### Step 5:
Now, integrating both terms:

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

We know that \(\int \tan^2(x) \, dx = \tan(x) - x + C\), so we have:

\[
x + (\tan(x) - x) + C
\]

### Step 6:
Simplifying the result gives:

\[
\tan(x) + C
\]

### Conclusion:
Yes, the solution in the image is **correctly solved**, and the final result is indeed:

\[
\tan(x) + C
\]

Would you like further details or clarifications?

### Related Questions:
1. How would the solution change if the integrand involved \(\sin^2(x)\) instead of \(\cos^2(x)\)?
2. What is the integral of \(\sec^2(x)\)?
3. How does using trigonometric identities simplify complex integrals?
4. Can we apply substitution methods for this integral?
5. How would this integral change if limits of integration were provided?

### Tip:
For integrals involving trigonometric functions, always check for identities like \(\cos(2x)\) and \(\tan^2(x)\) to simplify the problem.

Explore the step-by-step solution for the integral of (cos(2x) + sin^2(x))/cos^2(x) involving trigonometric identities like cos(2x) and tan^2(x). Learn how to simplify and solve integrals using key formulas and concepts in integral calculus. Suitable for advanced high school students.

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