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

To solve the integral

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

we can use a substitution method. This integral is directly related to the logarithmic integral and can be evaluated as follows:

### Step 1: Recognize the standard form

The integral

\[
\int \frac{\ln(x)}{x} \, dx
\]

is a known standard form whose solution is:

\[
\frac{\ln^2(x)}{2} + C
\]

### Step 2: Apply the limits of integration

Now, apply the limits \( x = 1 \) and \( x = 4 \):

\[
I = \left[ \frac{\ln^2(x)}{2} \right]_1^4
\]

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

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

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

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

### Step 3: Subtract the results

Thus, the value of the integral is:

\[
I = 2\ln^2(2) - 0 = 2\ln^2(2)
\]

So the final result is:

\[
\boxed{2\ln^2(2)}
\]

Let me know if you'd like further details or have questions! Here are five related questions to consider:

1. How does substitution work in logarithmic integrals?
2. Why does \( \ln(1) = 0 \)?
3. How do you handle integrals involving powers of logarithmic functions?
4. What is the difference between definite and indefinite integrals in general?
5. How would you evaluate integrals with limits involving non-integer logarithms?

**Tip:** Integrals involving logarithms often rely on recognizing patterns from standard forms, so keep a list of such forms handy!

Explore a detailed solution for the definite integral of ln(x)/x from 1 to 4. This problem covers logarithmic integration, definite integral evaluation, and the fundamental theorem of calculus. Suitable for high school or early college students.

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