2022-104

Consider the following integral:

$$\int {\cos }^{2}xdx$$

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