Question

If \( A = \int_0^{\infty} \frac{1 + x^2}{1 + x^4} dx \) and \( B = \int_0^1 \frac{1 + x^2}{1 + x^4} dx \), then:

• A. \( 2A = B \)
• B. \( A = B \)
• C. \( 2B = A \)
• D. \( 2B + A = 0 \)

Hint:

When dealing with integrals over symmetric intervals, use substitution and symmetry properties to relate the integrals to each other.

Answer 1

The Correct Option is C

We are given the integrals \[ A = \int_0^{\infty} \frac{1 + x^2}{1 + x^4} dx \quad \text{and} \quad B = \int_0^1 \frac{1 + x^2}{1 + x^4} dx. \] Step 1: Use symmetry properties of definite integrals and apply substitution to express \( A \) and \( B \) in terms of each other. This leads to the relation: \[ A = 2B. \] Thus, the correct answer is \( 2B = A \).