JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

Solution approach

1. Correct Answer + Core Reason: 3. The integral is evaluated by summing the integrals over integer intervals.
   
   Step-by-Step Solution:
   
   Step 1: Given $\int \nolimits_{\left[ x \right]}^{\left[ x \right] + 1} \,\,f\left( t \right)\,\,dt = \left[ x \right]$
   
   Step 2: Evaluate the integral from -2 to 4 by breaking it into intervals of length 1 based on the greatest integer function.
   
   Step 3: $\int \nolimits_{ - 2}^{ - 1} f\left( x \right)\,\,dx = -2$, $\int \nolimits_{ - 1}^0 \,\,f\left( x \right)\,\,dx = -1$, $\int \nolimits_0^1 \,\,f\left( x \right)\,\,dx = 0$, $\int \nolimits_1^2 \,\,f\left( x \right)\,\,dx = 1$, $\int \nolimits_2^3 \,\,f\left( x \right)\,\,dx = 2$, $\int \nolimits_3^4 \,\,f\left( x \right)\,\,dx = 3$
   
   Step 4: Sum the integrals: $\int \nolimits_{ - 2}^4 \,\,f\left( x \right)\,\,dx = -2 - 1 + 0 + 1 + 2 + 3 = 3$
   
   Why other options are wrong:
   - Option A: 1 is incorrect because it only considers a portion of the integral range or makes a calculation error.
   - Option B: 2 is incorrect because it only considers a portion of the integral range or makes a calculation error.
   - Option C: -2 is incorrect because it only considers a portion of the integral range or makes a calculation error.
   
   Key Concepts:
   - Definite Integrals
   - Greatest Integer Function
   - Properties of Integrals
   
   Memory Aids:
   - Remember to break the integral into intervals based on the greatest integer function.
   
   Quick Tips:
   - Always consider the properties of the greatest integer function when dealing with integrals involving it.