Final answer:
The integral of ln(x) / 2x dx is solved by using integration by parts, setting u = ln(x), and dv = (1/2x) dx. The result is (1/4) ln(x)^2, which requires substituting back into the original formula for u and v, and simplifying the result. The provided references to the exponential and natural logarithm relationships are not directly used in this calculation.
Step-by-step explanation:
The integral of ln(x) / 2x dx can be computed using the technique of integration by parts. Integration by parts is a method where the integral of a product of two functions is transformed into the integral of another product, based on the formula ∫ u dv = uv - ∫ v du. In this case, we let u = ln(x) and dv = (1/2x) dx. After differentiating and integrating appropriately, we apply the formula to solve for the integral.
First, compute the derivatives and integrals:
- Let u = ln(x), then du = (1/x) dx.
- Let dv = (1/2x) dx, then v = (1/2) ln(x).
Applying integration by parts:
- ∫ ln(x) / 2x dx = uv - ∫ v du
- = (1/2) ln(x)^2 - ∫ (1/2) ln(x) (1/x) dx
- = (1/2) ln(x)^2 - (1/2) ∫ (1/x) ln(x) dx
- = (1/2) ln(x)^2 - (1/4) [ln(x)^2]
- = (1/4) ln(x)^2 (As the constant of integration is arbitrary, it is omitted here)
This result represents the integral of the given function. The technique of expressing the number 2 as eln 2 is a mathematical trick not directly applicable in this integration process, but it does illustrate the relationship between exponents and logarithms.