Evaluate the Integral

Problem

(∫_^)(1/ln(x)*d(x))

Solution

1. Identify the integral as the logarithmic integral function, which is a special function in mathematics that cannot be expressed in terms of elementary functions.

2. Substitute u=ln(x) which implies x=eu and d(x)=eu*d(u)

3. Rewrite the integral in terms of u

(∫_^)((eu)/u*d(u))

4. Expand the term eu using its Taylor series representation:

eu=(∑_n=0^∞)((un)/(n!))

5. Divide the series by u and integrate term by term:

(∫_^)((1/u+(∑_n=1^∞)((u(n−1))/(n!)))*d(u))

6. Evaluate the resulting integral to find the series form:

ln(u)+(∑_n=1^∞)((un)/(n⋅n!))+C

7. Recognize that the standard notation for this specific integral is the logarithmic integral function, denoted as li(x)

Final Answer

(∫_^)(1/ln(x)*d(x))=li(x)+C

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