Sure, let's solve the integral [tex]\(\int_1^2 \frac{\ln(x)}{x} \, dx\)[/tex] step-by-step.
### Step-by-Step Solution
1. Identify the Integral and Recall a Relevant Technique:
[tex]\[
\int_1^2 \frac{\ln(x)}{x} \, dx
\][/tex]
We recognize that the integrand [tex]\(\frac{\ln(x)}{x}\)[/tex] suggests a possible use of integration techniques involving the natural logarithm function, especially because we know a specific property related to this integral.
2. Use a Known Property:
There's a well-known result that states for [tex]\( f(x) = \ln(x) \)[/tex], the integral [tex]\(\int \frac{\ln(x)}{x} \, dx\)[/tex] is a standard form. The antiderivative of this function is [tex]\(\frac{(\ln(x))^2}{2}\)[/tex].
3. Apply the Limits of Integration:
We need to evaluate this antiderivative from [tex]\(1\)[/tex] to [tex]\(2\)[/tex]:
[tex]\[
\left[ \frac{(\ln(x))^2}{2} \right]_1^2
\][/tex]
4. Evaluate the Antiderivative at the Boundaries:
[tex]\[
\left. \frac{(\ln(x))^2}{2} \right|_1^2 = \frac{(\ln(2))^2}{2} - \frac{(\ln(1))^2}{2}
\][/tex]
5. Simplify the Expression:
- Recall that [tex]\(\ln(1) = 0\)[/tex]:
[tex]\[
\frac{(\ln(1))^2}{2} = \frac{0^2}{2} = 0
\][/tex]
- Hence, the expression simplifies to:
[tex]\[
\frac{(\ln(2))^2}{2} - 0 = \frac{(\ln(2))^2}{2}
\][/tex]
### Final Answer
[tex]\[
\int_1^2 \frac{\ln(x)}{x} \, dx = \frac{(\ln(2))^2}{2}
\][/tex]
So, the evaluated integral is [tex]\(\boxed{\frac{(\ln(2))^2}{2}}\)[/tex].
