Calculate Integral Ln 2x Dx From 1 to 3

This guide explains how to calculate the definite integral of ln(2x) from 1 to 3. We'll cover the formula, step-by-step calculation, and interpretation of results.

How to Calculate the Integral

Calculating the definite integral of ln(2x) from 1 to 3 involves several steps. First, we need to find the antiderivative of ln(2x), then evaluate it at the upper and lower limits, and finally subtract the two results.

The Integral Calculation

The integral of ln(2x) with respect to x is calculated using integration by parts. The formula is:

∫ ln(2x) dx = x ln(2x) - ∫ (1/x) dx

This simplifies to x ln(2x) - ln|x| + C, where C is the constant of integration.

For definite integrals, we evaluate the antiderivative at the upper limit (3) and subtract its value at the lower limit (1).

The Formula

The general formula for the definite integral of ln(2x) from a to b is:

Definite Integral Formula

∫[a to b] ln(2x) dx = [x ln(2x) - ln|x|] evaluated from a to b

= [b ln(2b) - ln|b|] - [a ln(2a) - ln|a|]

For our specific case where a = 1 and b = 3:

Specific Calculation

∫[1 to 3] ln(2x) dx = [3 ln(6) - ln(3)] - [1 ln(2) - ln(1)]

= [3 ln(6) - ln(3)] - [ln(2) - 0]

= 3 ln(6) - ln(3) - ln(2)

Worked Example

Let's calculate the integral step by step:

Find the antiderivative: ∫ ln(2x) dx = x ln(2x) - ln|x|
Evaluate at upper limit (3): 3 ln(6) - ln(3)
Evaluate at lower limit (1): ln(2) - ln(1) = ln(2)
Subtract: [3 ln(6) - ln(3)] - ln(2)
Simplify: 3 ln(6) - ln(3) - ln(2)

Numerical Result

Calculating the numerical value:

3 ln(6) ≈ 3 × 1.7918 ≈ 5.3753

ln(3) ≈ 1.0986

ln(2) ≈ 0.6931

Final result ≈ 5.3753 - 1.0986 - 0.6931 ≈ 3.5836

Interpreting Results

The result of 3.5836 represents the area under the curve of ln(2x) between x=1 and x=3. This value is dimensionless since we're integrating a logarithmic function.

In practical terms, this integral might represent:

The accumulated growth of a process with logarithmic behavior
The total change in a quantity that grows logarithmically
A measure of the average logarithmic value over the interval

Practical Considerations

When interpreting results, remember that:

The integral gives the net accumulation
Negative values indicate net decrease
The result is context-dependent based on the original function

FAQ

What is the antiderivative of ln(2x)?

The antiderivative of ln(2x) is x ln(2x) - ln|x| + C, where C is the constant of integration.

Why do we need to use integration by parts for ln(2x)?

Integration by parts is required because ln(2x) is a product of logarithmic and linear functions, and there's no simple antiderivative for this combination.

What does the integral represent in real-world terms?

The integral represents the accumulated value of the logarithmic function over the specified interval, which could correspond to growth, change, or accumulation in various applications.

Can I calculate this integral with a calculator?

Yes, our calculator above performs this calculation automatically. You can also use scientific calculators or software like Wolfram Alpha for verification.