How do you evaluate the definite integral ∫ log x dx from [2, 4]?

To evaluate the definite integral ∫ log x dx from 2 to 4, we first need to find the antiderivative of log x. This can be done using integration by parts.

Step 1: Integration by Parts

For integration by parts, we use the formula:

∫ u dv = uv - ∫ v du

Let:

• u = log x (which means du = (1/x) dx)
• dv = dx (which means v = x)

Step 2: Applying the Formula

Now, substituting into the integration by parts formula:

∫ log x dx = x log x - ∫ x * (1/x) dx

This simplifies to:

∫ log x dx = x log x - ∫ dx

∫ log x dx = x log x - x + C

Step 3: Evaluate from 2 to 4

Now we can evaluate the definite integral:

∫ from 2 to 4 log x dx = [x log x - x] from 2 to 4

Calculating at the bounds:

At x = 4:

4 log 4 - 4 = 4(2) - 4 = 8 - 4 = 4

At x = 2:

2 log 2 - 2 = 2(0.693) - 2 ≈ 1.386 - 2 ≈ -0.614

Step 4: Final Calculation

Now, subtract the two results:

∫ from 2 to 4 log x dx = 4 - (-0.614) = 4 + 0.614 = 4.614

Result

The value of the definite integral ∫ log x dx from 2 to 4 is approximately 4.614.