Integration of log(logx)/x dx

To integrate the given function, log(logx)/x dx, we can use the technique of integration by substitution. Let's break down the integration process into steps:

Step 1: Identify the substitution variable
In this case, we can let u = logx. Taking the derivative of both sides gives du/dx = 1/x, which implies dx = x du. Now we can rewrite the integral using the substitution variable u.

Step 2: Rewrite the integral using the substitution variable
The given integral can be rewritten as ∫(log(u)/u) x du.

Step 3: Simplify the integral
We know that x du is equal to dx, so the integral becomes ∫log(u)/u dx.

Step 4: Solve the integral
To solve this integral, we can use integration by parts. This technique involves splitting the integral into two parts and applying a specific formula. The formula for integration by parts is:

∫u dv = uv - ∫v du

In this case, let's choose u = log(u) and dv = 1/u dx. Taking the derivatives and integrals, we have du = (1/u) du and v = ∫dx = x.

Step 5: Apply integration by parts
Using the integration by parts formula, the integral becomes:

∫log(u)/u dx = (log(u) * x) - ∫x * (1/u) du

Step 6: Simplify the integral
The integral ∫x * (1/u) du can be further simplified as ∫(x/u) du. We can rewrite x/u as (1/u^(-1)) to get ∫u^(-1) du.

Step 7: Solve the integral
Now, we can solve the integral ∫u^(-1) du. This is a standard integral and its solution is ln(u) + C, where C is the constant of integration.

Step 8: Substitute back the original variable
Substituting back u = log(x), we have ∫(log(x))^(-1) dx = ln(log(x)) + C.

Therefore, the integration of log(logx)/x dx is ln(log(x)) + C, where C is the constant of integration.