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The integral you've uploaded is:

\[
\int \left( \frac{1}{\sin^2 x} + \frac{1}{\cos^2 x} \right) \, dx
\]

Let's solve this step by step.

### Step 1: Use trigonometric identities
We know that:

\[
\frac{1}{\sin^2 x} = \csc^2 x \quad \text{and} \quad \frac{1}{\cos^2 x} = \sec^2 x
\]

So, the integral becomes:

\[
\int \left( \csc^2 x + \sec^2 x \right) \, dx
\]

### Step 2: Break the integral into two parts

\[
\int \csc^2 x \, dx + \int \sec^2 x \, dx
\]

### Step 3: Integrate each term

- The integral of \( \csc^2 x \) is \( -\cot x \).
- The integral of \( \sec^2 x \) is \( \tan x \).

Thus, the solution is:

\[
-\cot x + \tan x + C
\]

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

### Final Answer:

\[
\int \left( \frac{1}{\sin^2 x} + \frac{1}{\cos^2 x} \right) \, dx = -\cot x + \tan x + C
\]

Would you like further explanation on any of these steps?

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Here are 5 related questions to expand this topic:

1. How do you differentiate trigonometric functions like \( \csc^2 x \) and \( \sec^2 x \)?
2. Can you derive the integrals of \( \sec^2 x \) and \( \csc^2 x \)?
3. How would the result change if the integral was over limits, say from 0 to \( \frac{\pi}{2} \)?
4. What are the applications of integrals involving trigonometric functions in physics or engineering?
5. How does this integral relate to differential equations?

**Tip**: Always check if you can simplify integrals by applying known trigonometric identities before solving.

Learn how to solve the integral of (1/sin^2 x + 1/cos^2 x) step by step. This problem covers integral calculus and trigonometry concepts, including key formulas for integrating sec^2 x and csc^2 x. Suitable for high school students in Grades 11-12.

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