PYQs Based Test - 23 : Cauchy's Integral Theorem and Formula

Consider the integral ${\oint }_C\frac{\sin (x)}{x^2(x^2+4)}dx$

Where C is a counter-clockwise oriented circle defined as |x - i| = 2. The value of the integral is

The value of the integral

$\underset{-\infty }{\overset{\infty }{\int }}\frac{\sin x}{x^2+2x+2}dx$

evaluated using contour integration and the residue theorem is

C is a closed path in the z-plane given by |z| = 3. The value of the integral _C^ ≤ft( {{z^2 - z + 4j}z + 2j} )dz is

The value of ${\oint }_{\text{Γ}}^{}\frac{3z-5}{\left(z-1\right)\left(z-2\right)}dz$ along a closed path Γ is equal to (4 π i), where z = x + iy and i=√-1. The correct path Γ is

The value of the following complex integral, with C representing the unit circle centered at origin in the counterclockwise sense, is: ${\int }_C\frac{z^2+1}{z^2-2z}dz$