Integrals of complex functions and Integration by parts

In this class, we will deal only with integrals of complex functions of a real variable integrated with respected to the real variable. This is identical to integration of real functions of real variables. The $j$ is simply treated as a constant in all these cases. In this class, we are typically interested in time $t$ or frequency $\omega$ being the independent variable. Hence, the integrals will be with respect to $t$ or $\omega$.

For example,

$\int_0^{\pi/4} e^{j 2 t} \ dt = \left[\frac{e^{j 2 t}}{2 j}\right]_{0}^{\pi/4} = \frac{j-1}{2j}$

Sometimes we will have to use Integration by parts to evaluate integrals. The main result to recall is

(1)
\begin{align} \displaystyle{\int_a^b u(t) \ dv(t) \ dt = \left[ u(t) \ v(t) \right]_a^b - \int_a^b v(t) \ du(t) \ dt} \end{align}

Example:

Evaluate $\displaystyle{\int_0^1 t e^{-j \omega t} dt}$, where $\omega$ is any complex number
We choose

(2)
\begin{eqnarray} \nonumber u(t) = t &,& dv(t) = e^{-j \omega t} \ dt \\ \nonumber \Longrightarrow du(t)=dt &,& v(t) = \frac{e^{-j \omega t}}{-j \omega} \end{eqnarray}
(3)
\begin{eqnarray} \nonumber \int_0^1 t e^{-j \omega t} dt & = & \displaystyle{\left[\frac{t e^{-j \omega t}}{-j \omega}\right]_0^1 - \int_0^1 \frac{e^{-j \omega t}}{-j \omega} \ dt}\\ \nonumber & = & - e^{-j \omega} - \displaystyle{\left[\frac{e^{-j \omega t}}{-\omega^2} \right]_0^1} = - e^{-j \omega} + \frac{e^{-j \omega}-1}{\omega^2} \end{eqnarray}