Statement:
Let I = ∫[0, π/2] (sin x / x) dx

Solution:
To determine the interval in which I lies, we can evaluate the integral and analyze its value.

Step 1: Integrating the function:
To evaluate the integral, we can use the Taylor series expansion of sin x:

sin x = x - (x^3 / 3!) + (x^5 / 5!) - (x^7 / 7!) + ...

Therefore, the integral becomes:

I = ∫[0, π/2] (sin x / x) dx
= ∫[0, π/2] (x - (x^3 / 3!) + (x^5 / 5!) - (x^7 / 7!) + ...) / x dx

Simplifying the expression, we get:

I = ∫[0, π/2] (1 - (x^2 / 3!) + (x^4 / 5!) - (x^6 / 7!) + ...) dx

Step 2: Analyzing the integral:
Since sin x is an alternating function, the terms in the expansion of sin x / x alternate in sign. As x approaches 0, all the terms in the expansion become 0, except the first term (1).

As x approaches π/2, the terms in the expansion become smaller and smaller, converging towards 0.

Therefore, the integral I is positive and bounded by the limits of integration [0, π/2]. This means I lies in the interval [0, A], where A is the value of the integral evaluated at π/2.

Step 3: Evaluating the integral:
Let's evaluate the value of the integral at π/2:

I = ∫[0, π/2] (1 - (x^2 / 3!) + (x^4 / 5!) - (x^6 / 7!) + ...) dx
= 1 - (π^2 / 2^3) + (π^4 / 2^5*3!) - (π^6 / 2^7*5!) + ...

As the terms in the expansion decrease rapidly, we can approximate the value of I by taking a few terms:

I ≈ 1 - (π^2 / 2^3) ≈ 0.945

Conclusion:
Based on our analysis and evaluation of the integral, I lies in the interval [0, 0.945]. Therefore, the correct option is (b) [0, 1].