To find the indefinite integral of [tex]\(\csc(7x)\)[/tex], we need to follow a series of steps. Here's a step-by-step guide:
1. Recognize the Integral Form:
We need to integrate [tex]\(\csc(7x)\)[/tex] with respect to [tex]\(x\)[/tex].
[tex]\[
\int \csc(7x) \, dx
\][/tex]
2. Substitute for Simplicity:
Let [tex]\(u = 7x\)[/tex]. Thus, [tex]\(du = 7 \, dx\)[/tex] or [tex]\(\frac{du}{7} = dx\)[/tex].
3. Rewrite the Integral:
Substituting [tex]\(u\)[/tex] and [tex]\(dx\)[/tex] in terms of [tex]\(du\)[/tex]:
[tex]\[
\int \csc(7x) \, dx = \int \csc(u) \cdot \frac{1}{7} \, du
\][/tex]
4. Factor out the Constant:
The constant [tex]\(\frac{1}{7}\)[/tex] can be factored out of the integral:
[tex]\[
\frac{1}{7} \int \csc(u) \, du
\][/tex]
5. Integrate [tex]\(\csc(u)\)[/tex]:
The integral of [tex]\(\csc(u)\)[/tex] is known:
[tex]\[
\int \csc(u) \, du = \ln \left| \csc(u) - \cot(u) \right| + C
\][/tex]
6. Substitute Back [tex]\(u = 7x\)[/tex]:
Replace [tex]\(u\)[/tex] with [tex]\(7x\)[/tex] in the equation:
[tex]\[
\int \csc(7x) \, dx = \frac{1}{7} \ln \left| \csc(7x) - \cot(7x) \right| + C
\][/tex]
However, this integral can be simplified differently. Given our earlier completion and the known simplified result, the answer can be presented as:
[tex]\[
\boxed{\int \csc(7x) \, dx = \frac{1}{14} \ln \left| \frac{\cos(7x) - 1}{\cos(7x) + 1} \right| + C}
\][/tex]
Thus, the final solution is:
[tex]\[
\boxed{\int \csc(7x) \, dx = \frac{1}{14} \ln \left| \cos(7x) - 1 \right| - \frac{1}{14} \ln \left| \cos(7x) + 1 \right| + C }
\][/tex]
Or more compactly:
[tex]\[
\ln \left| \frac{\cos(7x) - 1}{\cos(7x) + 1} \right| / 14 + C
\][/tex]
This gives the same result as:
[tex]\[
C + \frac{1}{14} \ln |\cos(7x) - 1| - \frac{1}{14} \ln |\cos(7x) + 1|
\][/tex]
Therefore, the integral is:
[tex]\[
\boxed{\log \left| \frac{\cos(7x) - 1}{\cos(7x) + 1} \right| / 14 + C}
\][/tex]
