Correct Answer - Option 4 : i
The correct answer is option 4.
Calculation:
\(\displaystyle\int^{\pi/2}_0{\dfrac{\cos x+i\sin x}{\cos x-i \sin x}}dx\)
According to Euler’s formula, \(e^{iθ} = cos θ + i sin θ \\ e^{-iθ} = cos θ - i sin θ \)
Therefore
\(e^{ix} = cos x+ isinx \\ e^{-iθ} = cos θ - i sin θ \\ I=\int_{0}^π{ e^{ix} \over e^{-ix} } dx \\ I =\int_{0}^{π \over 2}{ e^{i2x} } dx\)
\( \\ I = [{ e^{i2x} \over 2i}]_{0}^{π \over 2} \\ I= {1 \over 2i}[cos2x+isin2x ]_{0}^{π \over 2} \\ I= {1 \over 2i} \times (-2) \\I={ -1 \over i } \\I=i \space \space \space \space \space \space \space \space (∵ {1 \over i} =-i) \)
Hence the correct answer is i.
