evaluate the indefinite integral of (5dx)/(xln(7x))

To evaluate the indefinite integral of (5dx)/(xln(7x)), we can use a method called substitution. Here's how you can do it:

Let's start by making a substitution. Let u = ln(7x).
To find the derivative of u with respect to x, we need to use the Chain Rule. The derivative of ln(7x) with respect to x is (1/x)(7), which simplifies to 7/x.

Now, solve for dx in terms of du:
dx = du / (7/x)

Next, let's substitute the value of dx and u into the integral:
∫ (5dx) / (xln(7x)) = ∫ (5 * du / (7/x)) / u
= (5/7) * ∫ (x/u) du

Now, we can simplify the expression by rearranging:
(5/7) * ∫ (x/u) du
= (5/7) * ∫ (x * (1/u)) du
= (5/7) * ∫ (x/u) du
= (5/7) * ∫ (x/u) du

Let's now integrate the expression ∫ (x/u) du:
∫ (x/u) du = x ∫ (1/u) du

To integrate 1/u with respect to u, we can use the natural logarithm. The integral of (1/u) du is ln|u| + C, where C is the constant of integration.

∫ (x/u) du = x ∫ (1/u) du = x * ln|u| + C

Finally, substitute back the value of u:
= x * ln|ln(7x)| + C

Therefore, the indefinite integral of (5dx)/(xln(7x)) is x * ln|ln(7x)| + C, where C is the constant of integration.