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The integral provided is:

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

First, we can simplify the expression inside the integral. Using the Pythagorean identity:

\[
\sin^2(x) + \cos^2(x) = 1
\]

So, the integral becomes:

\[
\int \frac{1}{\csc(x)} \, dx
\]

Since \(\csc(x) = \frac{1}{\sin(x)}\), we have:

\[
\int \sin(x) \, dx
\]

The integral of \(\sin(x)\) is:

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

Thus, the final result of the integral is:

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

Would you like further details or have any other questions? Here are 5 related questions for further practice:

1. How would you integrate \(\int \csc(x) \, dx\)?
2. What are the steps to integrate \(\int \sin^2(x) \, dx\)?
3. How does the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\) help simplify integrals?
4. How do you find the derivative of \(\cos(x)\)?
5. What is the antiderivative of \(\cos(x)\)?

**Tip:** Always look for trigonometric identities to simplify expressions before integrating.

Learn how to solve the integral ∫ (sen²(x) + cos²(x)) / csc(x) dx by simplifying the expression using the Pythagorean identity and basic trigonometric principles. This step-by-step solution is perfect for advanced high school students studying calculus and trigonometry.

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