Evaluate the indefinite integral.

∫4dx/(xln(x))

∫4dx/(xln(x))

Let u = lnx
Then du = dx/x and you have

∫ 4/u du

piece of cake, no?

To evaluate the indefinite integral ∫4dx/(xln(x)), we can use a substitution method. Let's use the substitution u = ln(x).

To find du, we will differentiate both sides of the substitution equation:

du/dx = 1/x [Differentiating both sides of u = ln(x)]

Now, we can rearrange the equation to solve for dx:

dx = x du

Substituting the value of dx in the integral, we get:

∫4dx/(xln(x)) = ∫4(x du)/(x u)

Simplifying the expression, we can cancel out the "x" terms:

∫4dx/(xln(x)) = ∫4 du/u = 4 ∫du/u

Now, we integrate ∫du/u, which is a standard integral:

∫ du/u = ln|u| + C

Finally, substituting the value of u back into the equation, we get:

∫4dx/(xln(x)) = 4ln|ln(x)| + C

Therefore, the evaluated indefinite integral of ∫4dx/(xln(x)) is 4ln|ln(x)| + C, where C is the constant of integration.