Let $\gamma$ be the path$$\gamma:\left[0,1\right]\rightarrow\mathbb{C}, t\rightarrow\exp\left(t+it\right)$$
I have found that $$\gamma'\left(t\right)=\left(1+i\right)\exp\left(t+it\right)$$
To find the length of this path, I need to find $\left|\gamma'\left(t\right)\right|$ but I'm not sure how to find the modulus.
I understand $\left|x+iy\right|=\sqrt{x^2+y^2}$ but I don't know how to apply this to this paticular function.
Use that
$$\exp (z) = \exp (\Re z)(\cos (\Im z) +i\sin (\Im z)).$$