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!