To evaluate the integral $\int \ln(10x) \, dx,$ we can use integration by parts. Let’s proceed by setting up the integration by parts formula.

Step-by-Step Solution

1. Choose $u$ and $dv$: For integration by parts, we set: $u = \ln(10x) \quad \text{and} \quad dv = dx.$

2. Differentiate $u$ and integrate $dv$:

   • $u = \ln(10x) \Rightarrow du = \frac{1}{10x} \cdot 10 \, dx = \frac{1}{x} \, dx$.
   • $dv = dx \Rightarrow v = x$.

3. Apply the integration by parts formula: The formula for integration by parts is: $\int u \, dv = uv - \int v \, du.$ Substituting in our values: $\int \ln(10x) \, dx = x \ln(10x) - \int x \cdot \frac{1}{x} \, dx.$

4. Simplify the remaining integral: $= x \ln(10x) - \int 1 \, dx.$ Now, integrate $\int 1 \, dx$: $= x \ln(10x) - x + C.$

Final Answer

$\int \ln(10x) \, dx = x \ln(10x) - x + C.$

Would you like further details on any part of this solution, or do you have additional questions?

Here are 5 related questions for practice:

1. How would you integrate $\int \ln(5x) \, dx$?
2. What is the result of $\int \ln(x^2) \, dx$?
3. How would the result change if we had $\int x \ln(10x) \, dx$?
4. Can you use a substitution method to evaluate $\int \ln(ax) \, dx$?
5. How does integration by parts work for $\int x \ln(x) \, dx$?

Tip: When integrating functions involving logarithms, integration by parts is often helpful, as it simplifies the logarithmic expression through differentiation.