The integral 

\[
\int \frac{1}{\sin(7x)} \, dx
\]

can be rewritten as:

\[
\int \csc(7x) \, dx.
\]

To solve this, we use a standard technique that involves a substitution to handle the form of the cosecant function.

### Step 1: Use Substitution

Let:

\[
u = 7x \implies du = 7 \, dx \implies dx = \frac{du}{7}.
\]

Substituting into the integral:

\[
\int \csc(7x) \, dx = \int \csc(u) \cdot \frac{du}{7} = \frac{1}{7} \int \csc(u) \, du.
\]

### Step 2: Integrate \(\csc(u)\)

The integral of \(\csc(u)\) is a standard result:

\[
\int \csc(u) \, du = -\ln |\csc(u) + \cot(u)| + C.
\]

So, substituting back:

\[
\frac{1}{7} \int \csc(u) \, du = -\frac{1}{7} \ln |\csc(u) + \cot(u)| + C.
\]

### Step 3: Substitute \(u = 7x\)

Now replace \( u \) with \( 7x \):

\[
-\frac{1}{7} \ln |\csc(7x) + \cot(7x)| + C.
\]

### Final Answer:

\[
\int \csc(7x) \, dx = -\frac{1}{7} \ln |\csc(7x) + \cot(7x)| + C.
\]

Would you like further clarification on any step, or do you have any other questions?

### Related Questions:
1. How do you evaluate \(\int \csc(x) \, dx\)?
2. Can you solve \(\int \csc^2(7x) \, dx\)?
3. What is the derivative of \(\csc(7x)\)?
4. How does substitution help in solving trigonometric integrals?
5. Can you find \(\int \sec(7x) \, dx\)?

### Tip:
Remember that integrals involving \(\csc\) or \(\sec\) often involve logarithmic forms, so be familiar with these standard results!

Learn how to solve the integral of 1 over sin(7x) step-by-step. This problem involves trigonometric integration and substitution techniques. Suitable for advanced high school and college students.

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