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:

Recognizing the symmetry of the integration limits and the parity (even or odd) of the integrand can significantly simplify definite integrals.

Answer 1

The Correct Option is C

Step 1: Check if the integrand \( f(x) = \frac{x + x^3 + x^5{1 + x^2 + x^4 + x^6} \) is odd or even.}
\[ f(-x) = \frac{(-x) + (-x)^3 + (-x)^5}{1 + (-x)^2 + (-x)^4 + (-x)^6} = \frac{-x - x^3 - x^5}{1 + x^2 + x^4 + x^6} = -f(x) \] Since \( f(-x) = -f(x) \), \( f(x) \) is an odd function.
Step 2: Use the property of definite integrals of odd functions over symmetric intervals.
If \( f(x) \) is odd, then \( \int_{-a}^{a} f(x) dx = 0 \).
Step 3: Apply the property to the given integral.
Here, \( a = 100 \), and \( f(x) \) is odd. \[ \int_{-100}^{100} \frac{x + x^3 + x^5}{1 + x^2 + x^4 + x^6} dx = 0 \]