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Evaluate the complex contour integral $$\int \limits_C Log(z) dz$$ where $Log(z)$ denotes the principal complex logarithm and $C$ is the straight line from $z=1 +i$ to $z= 3+ 3i$

My attempt:

We can parametrize the path $C$ as \begin{align}x &= t \\ y &= t \\ \therefore z(t) &= t + it \end{align} where $1 \le t \le 3$

Thus our integral can be written as \begin{align}\int \limits_1^3 \bigg[Log(t + it)\cdot(1 + i)\bigg] dt &= \int\limits_1^3 \bigg[Log(t + it)\bigg]dt + i\int \limits_1^3\bigg[Log(t + it)\bigg]dt\end{align}

Is this correct?

Also, am I correct in assuming that I must continue solving these two integrals using IBP by letting $u = Log(t + it)$ and $dv = dt$?

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You parametrization and the following identity are right, but there is a more efficient parametrization of $C$, given by $\varphi:[\sqrt{2},3\sqrt{2}]\mapsto t\cdot e^{\pi i/4}$, from which:

$$ \int_C \text{Log}(z)\,dz =e^{\pi i/4}\int_{\sqrt{2}}^{3\sqrt{2}}\left(\frac{\pi i}{4}+\log(t)\right)\,dt=e^{\pi i/4}\left(\frac{\pi i}{\sqrt{2}}-2\sqrt{2}+\sqrt{2}\log(2)+3\sqrt{2}\log(3)\right).$$

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