2022-104

Consider the following integral:

$$ \int \cos^2 x dx $$

Which of the alternatives has a correct intermediate step and the solution, using Euler's formulas below, where \( i \) represents the imaginary unit?

$$ \cos \theta = \frac{e^{\theta i} + e^{-\theta i}}{2} $$ $$ \sin \theta = \frac{e^{\theta i} - e^{-\theta i}}{2i} $$

1. $$ \frac{1}{2} \int (e^{2ix} + 2 - e^{-2ix}) dx = \frac{1}{2}(\frac{e^{2ix}}{2i} + 2x - \frac{e^{-2ix}}{2i}) + c $$
2. $$ \frac{1}{4} \int (e^{2ix} + 2 + e^{-2ix}) dx = \frac{1}{4}(\frac{e^{2ix}}{2i} + 2x - \frac{e^{-2ix}}{2i}) + c $$
3. $$ \frac{1}{4} \int (e^{2ix} + e^{-2ix}) dx = \frac{1}{4}(\frac{e^{2ix}}{2i} + \frac{e^{-2ix}}{2i}) + c $$
4. $$ \frac{1}{4} \int (e^{2x} + 2 + e^{-2x}) dx = \frac{1}{4}(\frac{e^{2x}}{2} + 2x + \frac{e^{-2x}}{2}) + c $$
5. None of the above
   

Original idea by: Diogo Souza