Question

The value of \[ \int_{-100}^{100} \frac{x+x^3+x^5}{1+x^2+x^4+x^6}\,dx \] is:

• A. \( 100 \)
• B. \( 1000 \)
• C. \( 0 \)
• D. \( 10 \)

Hint:

Always check parity: Odd / symmetric limits ⇒ zero.

Answer 1

The Correct Option is C

Concept: Check symmetry of integrand. Step 1: Check odd/even nature.} Numerator: \[ x+x^3+x^5 \quad \text{odd} \] Denominator: \[ 1+x^2+x^4+x^6 \quad \text{even} \] So integrand is odd. Step 2: Use symmetry.} Integral of odd function over symmetric limits: \[ [-a,a] \Rightarrow 0 \]