Correct Answer - Option 1 : 2 log 2 - 1

**Concept:**

**Integration by Parts:**

∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ [f'(x) ∫ g(x) dx] dx.

**Definite Integral:**

If ∫ f(x) dx = g(x) + C, then \(\rm ∫_a^b f(x)\ dx = [ g(x)]_a^b\) = g(b) - g(a).

**Calculation:**

Let's first integrate the expression under integral by parts.

I = ∫ log x dx = ∫ (1)(log x) dx

Considering log x as the first function and 1 as the second function, we get:

= (log x) ∫ 1 dx - ∫ [\(\rm\frac1x\) ∫ 1 dx] dx

= (log x) x - x + C

Putting the limits of the definite integral, we get:

\(\rm ∫_1^2 \log x \ dx\)

= \(\rm \left[x(\log x)-x\right]_1^2\)

= (2 log 2 - 2) - (0 - 1)

= **2 log 2 - 1**.