Question

The value of the integral \( \int_{2}^{4} \left( \frac{\log t}{t} \right) \, dt \) is equal to:

• A. \( \frac{1}{2}(\log 2)^2 \)
• B. \( \frac{5}{2}(\log 2)^2 \)
• C. \( \frac{3}{2}(\log 2)^2 \)
• D. \( (\log 2)^2 \)
• E. \( \frac{3}{2}(\log 2) \)

Hint:

Using properties of logarithms like $\log(a^n) = n\log a$ before or during evaluation makes handling the limits much simpler.

Answer 1

The Correct Option is C

Concept: This is another substitution problem where \( u = \log t \) and \( du = \frac{1}{t} \, dt \).

Step 1: Apply substitution.
Let \( u = \log t \). Then \( du = \frac{1}{t} \, dt \). Change the limits:
• When \( t = 2 \), \( u = \log 2 \).
• When \( t = 4 \), \( u = \log 4 = 2\log 2 \).

Step 2: Integrate.
\[ \int_{\log 2}^{2\log 2} u \, du = \left[ \frac{u^2}{2} \right]_{\log 2}^{2\log 2} \] \[ = \frac{1}{2} [(2\log 2)^2 - (\log 2)^2] \]

Step 3: Simplify.
\[ = \frac{1}{2} [4(\log 2)^2 - (\log 2)^2] \] \[ = \frac{1}{2} [3(\log 2)^2] = \frac{3}{2}(\log 2)^2 \]