Multiple choice

Given $i = \sqrt{-1}$, what will be the evaluation of the definite integral $\int \limits_0^{\pi/2} \frac{\cos x +i \sin x} {\cos x - i \sin x} dx$?

  1. 0

  2. 2

  3. – i

  4. i
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D Correct answer
Explanation

$\int_0^{\pi/2} \dfrac{e^{ix}}{e^{-ix}}dx = \int_0^{\pi/2} e^{2ix} dx$ $\left( \dfrac{e^{2ix}}{2i}\right) = \dfrac{1}{2i}[e^{ix}-1] = \dfrac{1}{2i}[\cos \pi + i \sin \pi -1] =\dfrac{1}{2i}[-1 + 0 -1] = \dfrac{-2}{2i} = \dfrac{-1}{i} \times \dfrac{i}{i} = \dfrac{-i}{-1} = i $

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