Questions: Evaluate the indefinite integral. [ int frac6 dxx ln (7x) ] +C

$Evaluate the indefinite integral. [ int frac6 dxx ln (7x) ] +C$

Transcript text: Evaluate the indefinite integral. \[ \int \frac{6 d x}{x \ln (7 x)} \] $\square$ $+C$

Solution

Solution Steps

To evaluate the indefinite integral $\int \frac{6 \, dx}{x \ln(7x)}$, we can use the method of substitution. Let $ u = \ln(7x) $, then $ du = \frac{1}{x} \, dx $. This substitution simplifies the integral into a form that is easier to integrate.

Step 1: Set Up the Integral

We start with the indefinite integral: \[ \int \frac{6 \, dx}{x \ln(7x)} \]

Step 2: Use Substitution

Let $ u = \ln(7x) $. Then, the differential $ du $ is given by: \[ du = \frac{1}{x} \, dx \quad \Rightarrow \quad dx = x \, du \] Since $ x = \frac{e^u}{7} $, we can substitute $ x $ back into the integral.

Step 3: Rewrite the Integral

Substituting $ u $ into the integral, we have: \[ \int \frac{6 \, dx}{x \ln(7x)} = \int \frac{6}{\frac{e^u}{7} \cdot u} \cdot \frac{e^u}{7} \, du = \int \frac{6 \cdot 7}{u} \, du = 6 \ln|u| + C \]

Step 4: Substitute Back

Substituting back for $ u $: \[ 6 \ln|\ln(7x)| + C \]