Question

Evaluate the integral: \[ \int_{-1}^{1} \left[ \sqrt{1 + x + x^2} - \sqrt{1 - x + x^2} \right] dx \]

• A. 2
• B. 5
• C. 1
• D. 0

Hint:

Look for symmetry in integrals to simplify the evaluation. In this case, symmetry about the origin led to the cancellation of terms.

Answer 1

The Correct Option is D

Step 1: Understanding the integral.
We are asked to evaluate the definite integral: \[ \int_{-1}^{1} \left[ \sqrt{1 + x + x^2} - \sqrt{1 - x + x^2} \right] dx \] Notice that the two integrand terms are symmetric in nature. Step 2: Symmetry of the integrand.
The terms \( \sqrt{1 + x + x^2} \) and \( \sqrt{1 - x + x^2} \) are symmetric about the origin. This suggests that the integral will evaluate to 0, as the contributions from \( x \) and \( -x \) will cancel each other out. Step 3: Conclusion.
Thus, the value of the integral is \(\boxed{0}\).