Question

Stoo proo Coslax) dx x(x++b%) Assuming that a and b are positive real number, solve the integral using "indented Contor"

Complex variables

Answer 1

The value of integration will be zero. Explanation: Use the substitution, As, real (elax) = cos ax Therefore, 00 | = reals elax x(x2+62) dx -00 Now_replace_x Z I = reals elaz z(z2+62) dz] -00 Now, 00 -00 1 = elaz z(z+ib)(z-ib) dz Now we have to solve the above integration by using contour method and then we will take only the real part of that integral value. Im(2) Z = ib 0 Re(z) =-ib The pole of the integration is z = 0, -ib and ib. According to the residue theorem, the value of integration is: 1 = 2ni [res (z = 0) + res (z = ib) + res (z = -ib)] But in the contour of semi-circle, the pole z= -ib lies out sides the contour, so res(z = -ib) = 0. Therefore, | = 2ni [res (z = 0) +res (z = ib)] Now,

Now, res (z = 0) = [z (zcz+ib*lz-ib))], 10-1660-ib) = ? (0+ib (0-ib) Z=0 res (z = ib) = [(z - ib) (zlz+ib)2-ib))], elib) ib(ib+ib) = 2ib(ib) e-ab -262 z=ib Hence, | = 2ni [52 + 1 = [e-ab 2). [Purely_complex_value] As we need the real part of above expression but from the above expression of I, it can be clearly observed that it is purely a complex value, so the real part of I is zero. Hence, the value of integration will be zero.