Question

Consider the integral \[ \int_C \frac{\sin(x)}{x^2(x^2 + 4)} \, dx \] where \( C \) is a counter-clockwise oriented circle defined as \( |x - i| = 2 \). The value of the integral is

• A. \( \frac{-\pi}{8} \sin(2i) \)
• B. \( \frac{\pi}{8} \sin(2i) \)
• C. \( \frac{-\pi}{4} \sin(2i) \)
• D. \( \frac{\pi}{4} \sin(2i) \)

Hint:

To compute contour integrals involving singularities, use the residue theorem. The integral is \( 2\pi i \times \) the sum of the residues inside the contour.

Answer 1

The Correct Option is C

We are given a contour integral where the contour \( C \) is a counter-clockwise oriented circle with a radius of 2 and centered at \( x = i \). To solve this, we use the residue theorem. The given integrand has singularities at the points where the denominator is zero, i.e., where \( x^2(x^2 + 4) = 0 \). These singularities are at \( x = 0 \) and \( x = \pm 2i \). Since \( C \) encloses the singularity at \( x = i \), we calculate the residue of the function at this point.
The residue at \( x = i \) can be computed by evaluating the function using the standard residue computation techniques. After applying the residue theorem, the result of the integral is \( \frac{-\pi}{4} \sin(2i) \).
Thus, the correct answer is option (C).
Final Answer: (C) \( \frac{-\pi}{4} \sin(2i) \)