To solve the integral $\int \frac{3 \, dx}{x \ln(4x)}$, we can follow these steps:

Step 1: Simplify the Integral

The integral is given by: $\int \frac{3 \, dx}{x \ln(4x)}$ We can factor out the constant $3$ from the integral: $3 \int \frac{dx}{x \ln(4x)}$

Step 2: Substitution

Let $u = \ln(4x)$. Now, differentiate $u$ with respect to $x$ to find $du$: $\frac{du}{dx} = \frac{1}{4x} \cdot 4 = \frac{1}{x}$ So, $du = \frac{dx}{x}$.

Step 3: Rewrite the Integral in Terms of $u$

Substitute $u$ and $du$ into the integral: $3 \int \frac{du}{u}$

Step 4: Integrate

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln|u|$: $3 \ln|u| + C$

Step 5: Substitute Back $u = \ln(4x)$

Replace $u$ with $\ln(4x)$: $3 \ln|\ln(4x)| + C$

Final Answer

The evaluated integral is: $\int \frac{3 \, dx}{x \ln(4x)} = 3 \ln|\ln(4x)| + C$

Would you like further details or have any questions about this process? Here are some related questions you might find helpful:

1. How do you approach substitution in integrals?
2. What are the conditions under which the natural logarithm is defined?
3. How does the constant factor influence the result of an integral?
4. What is the significance of the absolute value in the logarithm function?
5. Can this method be generalized to integrals of similar forms?

Tip: Always check if a substitution will simplify the integrand to a standard form before proceeding with more complex methods.