Question

evaluate the indefinite integral.

$$\int \frac{4 d x}{x \ln (5 x)}$$

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Explanation:

Step1: Use the constant multiple rule

The constant multiple rule of integration states that \(\int kf(x)dx = k\int f(x)dx\) for a constant \(k\). Here \(k = 4\), so \(\int\frac{4dx}{x\ln(5x)}=4\int\frac{dx}{x\ln(5x)}\)

Step2: Use substitution

Let \(u=\ln(5x)\). Then, by the chain - rule, \(du=\frac{1}{5x}\times5dx=\frac{dx}{x}\)

Step3: Integrate the new function

Substituting \(u\) and \(du\) into the integral, we get \(4\int\frac{du}{u}\)
The integral of \(\frac{1}{u}\) with respect to \(u\) is \(\ln|u|+C\) (where \(C\) is the constant of integration). So \(4\int\frac{du}{u}=4\ln|u| + C\)

Step4: Substitute back for \(u\)

Since \(u = \ln(5x)\), the result is \(4\ln|\ln(5x)|+C\)

Answer:

\(4\ln|\ln(5x)|+C\)