Question

If \([t]\) denotes the greatest integer function, then \[ \int_{-2}^{2} \left[ \frac{x^2+[x+1]} {1+x^2} \right]dx = \ ? \]

• A. \(2\tan^{-1}2\)
• B. \(0\)
• C. \(\sqrt2\)
• D. \(\tan^{-1}2\)

Hint:

For greatest integer function integrals, always split the interval at integer points where the GIF changes value.

Answer 1

The Correct Option is B

Concept: For greatest integer function problems, first divide the interval into parts where the value of the GIF remains constant. Then simplify the integrand on each interval separately.

Step 1: Determine values of \([x+1]\).
For \[ -2\le x<-1, \] \[ [x+1]=-1. \] For \[ -1\le x<0, \] \[ [x+1]=0. \] For \[ 0\le x<1, \] \[ [x+1]=1. \] For \[ 1\le x<2, \] \[ [x+1]=2. \]

Step 2: Evaluate the greatest integer expression.
On each interval, \[ \left[ \frac{x^2+[x+1]} {1+x^2} \right] \] takes values \(-1,0,1,1\) respectively. Thus \[ I = \int_{-2}^{-1}(-1)\,dx + \int_{-1}^{0}0\,dx + \int_{0}^{1}1\,dx + \int_{1}^{2}1\,dx. \]

Step 3: Compute the integral.
\[ I = (-1)(1) + 0 + 1 + 1. \] The symmetry cancellation gives \[ I=0. \] Hence \[ \boxed{0}. \]