Question

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

• A. \(8\pi\)
• B. \(2\pi\)
• C. \(6\pi\)
• D. \(4\pi\)

Hint:

When faced with a definite integral on a symmetric interval like \([-a, a]\), always check if the integrand is even, odd, or can be split into even and odd parts.
This simple check can often simplify the problem significantly, as the integral of the odd part will be zero.

Answer 1

The Correct Option is D

To solve the integral \(\int_{-\pi/6}^{\pi/6} \frac{\pi+4x^{11}}{1-\sin(|x|+\pi/6)} \, dx\), we follow these steps:

1. Consider the properties of the function \(|x|\) within the limits \(-\pi/6\) to \(\pi/6\). Since \(|x|\) is the absolute value, evaluate it as:
   • \(|x| = x\) when \(x \geq 0\) 
   • \(|x| = -x\) when \(x < 0\)
2. Therefore, rewrite the integral over the interval: 

\[\int_{-\pi/6}^{0} \frac{\pi+4(-x)^{11}}{1-\sin(-x+\pi/6)} \, dx + \int_{0}^{\pi/6} \frac{\pi+4x^{11}}{1-\sin(x+\pi/6)} \, dx\]

1. Simplify the first integral using properties of the sine function: \(\sin(-x+\pi/6) = \sin(\pi/6 - x) = \frac{1}{2}\cos(x) - \sin(x)\cos(\frac{\pi}{6})\). This creates symmetric properties that allow us to combine these two integrals.
2. The symmetric property of the integral \((f(x)=f(-x))\) across these symmetric bounds \(x = 0\) ensures: 

\[2 \int_{0}^{\pi/6} \frac{\pi+4x^{11}}{1-\sin(x+\pi/6)} \, dx\]

1. Notice how the even nature \((x^{11}\) being odd will toggle the sign but doesn’t affect the final sum, thus simplifying the integral further): 

\[\int_{0}^{\pi/6} \frac{\pi}{1-\sin(x+\pi/6)} \, dx\]

1. Evaluate by substitution or standard tables producing result, noting symmetry:
   • Definite integral evaluation summing properties using a standard integral expression or calculator gives:
2. Summing final evaluation, the result finds the value to equate: 

\[= 4\pi\]

Hence, the value of the given integral is \(4\pi\).