Which calculus is Conway mentioning? I'm reading Conway's complex analysis book and on page 62 he proved this theorem:

Theorem 1.9. If $\gamma$ is piecewise smooth and $f:[a,b]\to \mathbb C$ is continuous then
  $$\int_a^bfd\gamma=\int_a^bf(t)\gamma'(t)dt$$

However he made the following claim before this theorem:

The following theorem says that in this case we can find $\int
 fd\gamma$ by the methods of integration learned in calculus.

These functions are complex valued ones, I only know how to integrate the real valued ones.
Which calculus is the author mentioning?
 A: For $f:[a,b]\to\Bbb C$ continuous, we write $f(t)=u(t)+i\,v(t)$ for real-valued function $u,v:[a,b]\to\Bbb R$ and define
$$\int_a^bf(t)dt:=\int_a^bu(t)dt+i\int_a^b v(t)dt.$$
A: The right hand side is an ordinary integral on the real interval $[a,b]$, at least if the real and imaginary parts of the integrand are considered separately.
That is, the real part of the integrand and the imaginary part may by linearity of the integral operator be combined to give a complex definite integral that is otherwise in accord with a Riemann or a Lebesgue definition.
Note that both $f(t)$ and $\gamma'(t)$, the "slope" of the path of integration, will in general contribute to these real and imaginary parts:
$$ f(t) \gamma'(t) = u(t) + i v(t) $$
where $u,v:[a,b] \to \mathbb{R}$ are continuous.
This would be straightforward if both the real and imaginary parts of the parameterized path $\gamma:[a,b] \to \mathbb{C}$ were smooth (differentiable).  In practice we often employ paths that are only piecewise-differentiable (but continuous), as when a "contour integral" is contrived with a keyhole or other convenient shape for avoiding singularities in the integrand $f$.
But piecewise-differentiable is good enough, since we then (typically without further discussion) break up the interval of integration $[a,b]$ into segments on which $\gamma$ is differentiable (in both real and imaginary components, as a function of $t$) and consider the definite integral as a sum of definite integrals on those segments that partition $[a,b]$.
