How to integrate ln x/x?

To find Integration of ln x/x, we will use the substitution rule for integration.

Answer: The final integral of ln x/x is (1/2) ln(x)² + c.

Go through the explanation to understand better.

Explanation:

Given:

y = ln x/x

y = (1/x)ln x

Now, we have the two functions;

f(x) = 1/x --------------- (1)

g(x) = ln(x) ------------- (2)

We know that the derivative of ln(x) is (1/x), so f(x) = g'(x).

We have to use integration by substitution to solve the original equation.

Let u = ln(x) ------------- (3)

du/dx = 1/x

du = (1/x)dx --------------- (4)

Substituting from equations (1), (2), (3) and (4) in the original integral we get,

∫ ln(x) (1/x) dx = ∫u du = (1/2) u² + c

Re-substituting for u gives us;

= (1/2) ln(x)² + c

Thus, the final integral of ln x/x is (1/2) ln(x)² + c.