Evaluate the indefinite integral of (5dx)/(xln(6x))
To evaluate the indefinite integral of (5dx)/(xln(6x)), we can use the method of integration by substitution.
Let's start by making a substitution. Let u = ln(6x), then du = (1/x) dx.
Now, we can rewrite the integral in terms of u:
∫(5dx)/(xln(6x)) = ∫(5dx)/(xu)
Next, substitute du = (1/x) dx into the integral:
∫(5dx)/(xln(6x)) = ∫(5du)/u
Integrating (5du)/u is a straightforward process:
∫(5du)/u = 5∫(du)/u = 5ln|u| + C
Finally, substitute back u = ln(6x) into the expression:
5ln|u| + C = 5ln|ln(6x)| + C
Therefore, the indefinite integral of (5dx)/(xln(6x)) is 5ln|ln(6x)| + C, where C is the constant of integration.