Question

The value of \[ \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \frac{\pi + 4x^{11}}{1 - \sin\left( \left| x \right| + \frac{\pi}{6} \right)} dx \] is

• A. \( 3\pi \)
• B. \( 4\pi \)
• C. \( 6\pi \)
• D. \( 12\pi \)

Hint:

For integrals involving symmetry, split the integral into even and odd parts to simplify the calculation.

Answer 1

The Correct Option is B

Step 1: Simplify the integral.
We are given the integral: \[ \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \frac{\pi + 4x^{11}}{1 - \sin\left( \left| x \right| + \frac{\pi}{6} \right)} dx \] The function is symmetric around 0 because the limits of integration are from \( -\frac{\pi}{6} \) to \( \frac{\pi}{6} \). Therefore, the integral can be simplified by considering the symmetry of the integrand. Step 2: Use symmetry properties.
By splitting the integral into two parts, we notice that the odd function \( 4x^{11} \) will contribute 0 to the integral, so we are left with the even function: \[ \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \frac{\pi}{1 - \sin\left( \left| x \right| + \frac{\pi}{6} \right)} dx \] Step 3: Evaluate the integral.
The integral can be computed directly by using standard integral techniques, and the final result is \( 4\pi \).