Definite Complex Integral

Problem 137: Evaluate the following complex integral

(1) $\begin{equation*} \int_{0}^{1} (1+it)^2 \ dt.\end{equation*}$

Solution: When evaluating complex integral, we consider the real part and imaginary part of the integral separately. That is,

(2) $\begin{align*} \int_{0}^{1} (1+it)^2 \ dt & = \int_{0}^{1} 1-t^2 + 2it \ dt \\ & = \underbrace{\int_{0}^{1} 1-t^2 \ dt}_{\text{real part}} + i \underbrace{\int_{0}^{1} 2t \ dt}_{\text{imaginary part}} \\ & = t - \frac{t^3}{3} \bigg|_{0}^{1} + i \left(\frac{2t^2}{2}\right)\bigg|_{0}^{1} \\ & = 1 - \frac{1}{3} + i \\ & = \frac{2}{3} + i. \end{align*}$

That is,

(3) $\begin{equation*} \int_{0}^{1} (1+it)^2 \ dt = \frac{2}{3} + i.\end{equation*}$

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