\int_{1}^{4}\frac{\ln\left(x\right)}{x}dx

The given integral is:

\[
\int_{1}^{4} \frac{\ln(x)}{x} \, dx
\]

This is a classic integral that can be solved using the technique of substitution. 

1. **Recall the formula**:

\[
\int \frac{\ln(x)}{x} \, dx = \frac{(\ln(x))^2}{2} + C
\]

Now, let's apply the limits of integration from 1 to 4.

2. **Evaluating the definite integral**:

\[
\int_{1}^{4} \frac{\ln(x)}{x} \, dx = \left[ \frac{(\ln(x))^2}{2} \right]_{1}^{4}
\]

3. **Substitute the limits**:

First, evaluate at the upper limit \( x = 4 \):

\[
\frac{(\ln(4))^2}{2} = \frac{(\ln(2^2))^2}{2} = \frac{(2\ln(2))^2}{2} = \frac{4(\ln(2))^2}{2} = 2(\ln(2))^2
\]

Now, evaluate at the lower limit \( x = 1 \):

\[
\frac{(\ln(1))^2}{2} = \frac{0^2}{2} = 0
\]

4. **Final result**:

\[
\int_{1}^{4} \frac{\ln(x)}{x} \, dx = 2(\ln(2))^2
\]

This is the exact value of the definite integral.

---

Would you like further details on any step? Here are 5 related questions for you to explore:

1. How can substitution be used to solve integrals with different logarithmic functions?
2. What is the significance of the natural logarithm in integration problems like this?
3. How do we derive the integral \(\int \frac{\ln(x)}{x} dx\)?
4. How does the method of integration by parts apply to similar integrals?
5. What are the properties of definite integrals over logarithmic functions?

**Tip:** Integrals involving logarithmic functions often simplify using properties of logarithms, such as \(\ln(a^b) = b\ln(a)\).

Learn how to solve the definite integral \int_{1}^{4} \frac{\ln(x)}{x} dx. This step-by-step solution involves the formula \int \frac{\ln(x)}{x} dx = \frac{(\ln(x))^2}{2} and applies the limits of integration from 1 to 4. This problem is suitable for college-level calculus students.

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