How do you integrate ;

ln(lnx)

and x/lnx

To integrate ln(ln(x)), we can use integration by parts. This technique involves choosing two parts of the function and performing integration on one while differentiating the other. The general formula for integration by parts is:

∫ u dv = uv - ∫ v du

Let's take ln(ln(x)) as u and 1 as dv:

u = ln(ln(x)), dv = 1

Now, we differentiate u and integrate dv:

du = (1 / ln(x)) * (1/x), v = x

Using the integration by parts formula, we have:

∫ ln(ln(x)) dx = x * ln(ln(x)) - ∫ x * (1 / ln(x)) * (1/x) dx

Simplifying this expression, we get:

∫ ln(ln(x)) dx = x * ln(ln(x)) - ∫ (1 / ln(x)) dx

Now, let's focus on integrating x/ln(x).

To integrate x/ln(x), we can use a substitution technique. Let's make the substitution:

u = ln(x) => du = (1/x) dx

Replacing x and dx in terms of u, we have:

x = e^u, dx = e^u du

Substituting these values into x/ln(x):

∫ x/ln(x) dx = ∫ (e^u)/(u) e^u du

Simplifying the expression further:

∫ x/ln(x) dx = ∫ e^(2u) / u du

Unfortunately, this integral does not have a simple closed-form solution using elementary functions. It is known as the exponential integral Ei(x). Therefore, we cannot find an exact expression for the antiderivative of x/ln(x) using standard functions.