Integral Help!?

Question

Ok show how to solve the integral [(sin(z)(z^3 - 2z)] / (2z^5 - 3z)

Answer 1

There is no analytical expression for the indefinite integral. However I believe this is from a complex numbers class. It can only work for definite integrals, and I presume the limits are -inf to +inf. So I'll work that out, using the Residual theorem.

f(z) = e^iz * (z^2 - 2) / (2z^4 - 3)
It has poles at a, -a, ia, -ia
where a = (3/2)^(1/4)
Since e^iz = cos(z) + i*sin(z)
we are interested in the imaginary component of the integral.

The integral =
2π*i*∑Res(f,p in D) - π*i*∑Res(f, p on C)

The poles +a and -a are on C, while ia is in D.

So it's just a matter of working those residuals.

Integral = π/4 *e^(-a)* (a^2+2)/a^3 + π/4 *sin(a) *(a^2-2)/a^3

Since there is no imaginary component, the desired integral is zero. It makes sense since f(z) is an odd function, the area cancels off over -inf to +inf.
If you had cos(z) instead, the answer would have been the one above.