Answer
\[\frac{3}{2}\ln \left( \frac{3}{2} \right)\mathbf{i}-2\mathbf{k}\]
Work Step by Step
\[\begin{align}
& \int_{1/2}^{1}{\left( \frac{3}{1+2t}\mathbf{i}-\pi {{\csc }^{2}}\left( \frac{\pi }{2}t \right)\mathbf{k} \right)dt} \\
& \text{Integrate} \\
& =\left[ \frac{3}{2}\ln \left| 1+2t \right|\mathbf{i}+\pi \left( \frac{2}{\pi } \right)\cot \left( \frac{\pi }{2}t \right)\mathbf{k} \right]_{1/2}^{1} \\
& =\left[ \frac{3}{2}\ln \left| 1+2t \right|\mathbf{i}+2\cot \left( \frac{\pi }{2}t \right)\mathbf{k} \right]_{1/2}^{1} \\
& \text{Evaluate} \\
& =\left[ \frac{3}{2}\ln \left| 1+2\left( 1 \right) \right|\mathbf{i}+2\cot \left( \frac{\pi }{2}\left( 1 \right) \right)\mathbf{k} \right] \\
& -\left[ \frac{3}{2}\ln \left| 1+2\left( \frac{1}{2} \right) \right|\mathbf{i}+2\cot \left( \frac{\pi }{2}\left( \frac{1}{2} \right) \right)\mathbf{k} \right] \\
& \text{Simplifying} \\
& =\frac{3}{2}\ln \left| 3 \right|\mathbf{i}+2\cot \left( \frac{\pi }{2} \right)\mathbf{k}-\frac{3}{2}\ln \left| 2 \right|\mathbf{i}-2\cot \left( \frac{\pi }{4} \right)\mathbf{k} \\
& =\frac{3}{2}\ln 3\mathbf{i}+2\left( 0 \right)\mathbf{k}-\frac{3}{2}\ln 2\mathbf{i}-2\left( 1 \right)\mathbf{k} \\
& =\frac{3}{2}\ln 3\mathbf{i}-\frac{3}{2}\ln 2\mathbf{i}-2\mathbf{k} \\
& =\frac{3}{2}\ln \left( \frac{3}{2} \right)\mathbf{i}-2\mathbf{k} \\
\end{align}\]