$\sigma=[0,1]+[1,i]$ is a contour. I am asked to sketch the contour, and evaluate $\int_\sigma Re(z)$.
Firstly, I am not sure how to visualise this contour, since there are two parts. What does it look like?
Onto the integral: I know that the formula for integrating a contour $\gamma$ is $\int_\gamma f(z)dz=\int_a^bf(\gamma(t))\gamma'(t)$, where $[a,b]$ is an interval.
In the solutions, the integral has been split into two parts: $\sigma_1(t)=t$ ($0\le t\le1$) and $\sigma_2(t)=1+(i-1)t$ ($0\le t\le1$). What is the reason for splitting up $\sigma$, and how do we deduce the expressions for $\sigma_1$ and $\sigma_2$, as well as the interval $0\le t \le 1$?
From here on, I understand how to obtain the solution: $f(\sigma_1(t))=Re(\sigma_1(t))=t$, and $\sigma'(t)=1$. Hence, $\int_\gamma f(z)dz=\int_a^bf(\gamma(t))\gamma'(t)=\int_0^1t dt=\frac{1}{2}$. Similarly, $f(\sigma_2(t))=Re(\sigma_2(t))=1-t$, and $\sigma_2'(t)=i-1.$ Hence, $\int_0^1(1-t)(i-1)dt = \frac{i-1}{2}$.
To finish off, $\int_\sigma Re(z)dz = \int_{\sigma_1}Re(z) dz + \int_{\sigma_2}Re(z) dz$ = $\frac{i}{2}$.
So, really, I am only stuck on firstly visualising the contour, and secondly, the intuition behind separating $\sigma$ into $\sigma_1$ and $\sigma_2$. Any help would be very much appreciated.
Well, in the statement of your problem, it's not really clear what those two "closed intervals" mean. After you give the parameterization, it's clearer.
Here is a plot:
The contour is separated into two parameterizations because it consists of two line segments.