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To find the integral of the given function \( \csc(x) \cot(x) - \frac{\sqrt{x}}{\sqrt[3]{x}} \), we will integrate each term separately.

Given:
\[ f(x) = \csc(x) \cot(x) - \frac{\sqrt{x}}{\sqrt[3]{x}} \]

Let's break it down term by term.

1. **Integral of \( \csc(x) \cot(x) \)**:

We know that the derivative of \( \csc(x) \) is \( -\csc(x) \cot(x) \). Therefore, the integral of \( \csc(x) \cot(x) \) is:

\[ \int \csc(x) \cot(x) \, dx = -\csc(x) + C_1 \]

2. **Simplify \( \frac{\sqrt{x}}{\sqrt[3]{x}} \)**:

Express the roots as exponents:

\[ \sqrt{x} = x^{1/2} \]
\[ \sqrt[3]{x} = x^{1/3} \]

Thus, the term becomes:

\[ \frac{\sqrt{x}}{\sqrt[3]{x}} = \frac{x^{1/2}}{x^{1/3}} = x^{1/2 - 1/3} = x^{3/6 - 2/6} = x^{1/6} \]

3. **Integral of \( x^{1/6} \)**:

Use the power rule for integration:

\[ \int x^{1/6} \, dx = \frac{x^{1/6 + 1}}{1/6 + 1} = \frac{x^{7/6}}{7/6} = \frac{6}{7} x^{7/6} + C_2 \]

Combining both parts, we get the integral of the entire function:

\[ \int \left( \csc(x) \cot(x) - \frac{\sqrt{x}}{\sqrt[3]{x}} \right) dx \]

\[ = \int \csc(x) \cot(x) \, dx - \int \frac{\sqrt{x}}{\sqrt[3]{x}} \, dx \]

\[ = -\csc(x) - \frac{6}{7} x^{7/6} + C \]

Thus, the integral of \( f(x) \) is:

\[ \int f(x) \, dx = -\csc(x) - \frac{6}{7} x^{7/6} + C \]

Would you like more details or have any questions about this solution?

Here are some related questions you might find useful:

1. How do you differentiate and integrate trigonometric functions?
2. What are some common trigonometric identities used in calculus?
3. How do you handle integrals involving trigonometric functions with different arguments?
4. Can you provide an example of integrating a trigonometric function with a substitution method?
5. What is the relationship between the derivative and integral of \( \csc(x) \cot(x) \)?
6. How do you simplify expressions with roots before integrating?
7. How do you determine the appropriate integration technique for complex integrands?
8. What are some tips for checking the correctness of an integral solution?

**Tip:** When dealing with trigonometric integrals, remember the derivatives of the basic trigonometric functions as they often directly help in finding the integrals.

Discover how to find the integral of the function csc(x) cot(x) - sqrt(x) / cuberoot(x) step-by-step. This problem involves trigonometric functions, exponents, and the power rule for integration. Suitable for advanced high school students and calculus learners.

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