You provided an integral expression, but it seems unclear. Based on the notation, I interpret it as:

\[
\int \frac{\ln(a)}{x} \sin^{-1}(x) \, dx
\]

This involves the natural logarithm (\(\ln a\)) and the inverse sine (\(\sin^{-1}(x)\)). Here’s how we can approach it:

### Step-by-step Solution:

1. **Factor out constants**:
   Since \(\ln(a)\) is independent of \(x\), it can be factored out of the integral:
   \[
   \ln(a) \int \frac{1}{x} \sin^{-1}(x) \, dx
   \]

2. **Integration by parts**:
   To solve \(\int \frac{1}{x} \sin^{-1}(x) \, dx\), we use integration by parts.

   Let:
   - \(u = \sin^{-1}(x)\) so that \(du = \frac{1}{\sqrt{1 - x^2}} \, dx\)
   - \(dv = \frac{1}{x} \, dx\), which gives \(v = \ln|x|\)

   Now apply the integration by parts formula:
   \[
   \int u \, dv = uv - \int v \, du
   \]
   This gives:
   \[
   \ln|x| \sin^{-1}(x) - \int \ln|x| \cdot \frac{1}{\sqrt{1 - x^2}} \, dx
   \]

   The second integral is more complex and requires further techniques like substitution or series expansion for evaluation.

Let me know if you'd like to proceed with this integral in more detail or explore it further.

### Expanding on related topics:
1. Would you like to explore the solution to the remaining integral?
2. How does integration by parts apply to other inverse trigonometric functions?
3. What methods can be used to solve integrals involving logarithmic functions?
4. Do you want to explore the series expansion approach for complex integrals?
5. Would you like to examine specific trigonometric identities that might simplify this integral?

**Tip**: Always try factoring out constants before applying complex integration techniques, as it simplifies the problem and avoids unnecessary complications.

Explore the solution to the integral ln(a) / x * sin⁻¹(x) dx with a detailed step-by-step approach using integration by parts. This problem covers key calculus concepts, including the inverse sine function and logarithmic integration, suitable for undergraduate-level students.

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