Evaluate the definite integral: $2{\int }_{\frac{\pi }{3}}^{\frac{\pi }{3}}{csc}^2(\frac{1}{2}t)dt$.

• A. $2$
• B. $\frac{\pi }{3}$
• C. $1$
• D. $0$

Answer: (D)

Verified step by step guidance

1. Step 1: Recognize that the integral is a definite integral with the same upper and lower limits, π/3 and π/3. In general, for any definite integral where the upper and lower limits are the same, the integral evaluates to 0.
2. Step 2: Understand the mathematical reasoning behind this. The definite integral represents the net area under the curve of the function between the given limits. If the limits are identical, there is no interval to calculate the area, and thus the integral is 0.
3. Step 3: Note that the function inside the integral, \( \csc^2 \left( \frac{1}{2} t \right) \), does not affect the result because the limits of integration are the same.
4. Step 4: Confirm that this property of definite integrals applies universally, regardless of the function being integrated. This is a fundamental property of definite integrals.
5. Step 5: Conclude that the value of the given integral is 0 without performing any further calculations, as the integral's limits dictate the result.