This technique requires an understanding and recognition of complex numbers. Specifically Euler's formula:
![{\displaystyle \cos(\theta )+i\sin(\theta )=e^{\theta i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50d2990766f086334f4ea0a14616479cf6f6dacc)
Recognize, for example, that the real portion:
![{\displaystyle {\text{Re}}\left\{e^{\theta i}\right\}=\cos(\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e449fb3f63e907ad097c6d2455fb057c2729bc26)
Given an integral of the general form:
![{\displaystyle \int e^{x}\cos(2x)dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eaba1718dfe9c7d8d134b34c213a037d0c5c21c1)
We can complexify it:
![{\displaystyle \int {\text{Re}}{\Big \{}e^{x}{\big (}\cos(2x)+i\sin(2x){\big )}{\Big \}}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0561127ecab6e116e9209547bdd309843883cb9d)
![{\displaystyle \int {\text{Re}}{\big \{}e^{x}(e^{2xi}){\big \}}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a39760ded386fabc7d9f72b580bdeec7d447461)
With basic rules of exponents:
![{\displaystyle \int {\text{Re}}\{e^{x+2ix}\}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e37b0656ce1edeb0e6aea60627dc7972cfe2fe)
It can be proven that the "real portion" operator can be moved outside the integral:
![{\displaystyle {\text{Re}}\left\{\int e^{x(1+2i)}dx\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0579498d5f84f711b3b476782a0fb17c99fcdafc)
The integral easily evaluates:
![{\displaystyle {\text{Re}}\left\{{\frac {e^{x(1+2i)}}{1+2i}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c3ec95f9e592d502f1dc7a463f9d8691ce3a0c7)
Multiplying and dividing by
:
![{\displaystyle {\text{Re}}\left\{{\frac {1-2i}{5}}e^{x(1+2i)}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d857195a0bad302f4810b3ab50e33d764b2323)
Which can be rewritten as:
![{\displaystyle {\text{Re}}\left\{{\frac {1-2i}{5}}e^{x}e^{2ix}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37574d1bc4d15586aa8a3dcaa45513fae3844c8d)
Applying Euler's forumula:
![{\displaystyle {\text{Re}}\left\{{\frac {1-2i}{5}}e^{x}{\big (}\cos(2x)+i\sin(2x){\big )}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9e8e5c8a013a0bd24eb59a27c1f365953919a6)
Expanding:
![{\displaystyle {\text{Re}}\left\{{\frac {e^{x}}{5}}{\big (}\cos(2x)+2\sin(2x){\big )}+i\cdot {\frac {e^{x}}{5}}{\big (}\sin(2x)-2\cos(2x){\big )}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e98ac10f0e00f497de8cba140e94e977f009102b)
Taking the Real part of this expression:
![{\displaystyle {\frac {e^{x}}{5}}{\big (}\cos(2x)+2\sin(2x){\big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95fe209eca2ade5ee5d01c3237b2c15d34a75d29)
So:
![{\displaystyle \int e^{x}\cos(2x)dx={\frac {e^{x}}{5}}{\big (}\cos(2x)+2\sin(2x){\big )}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3c5761027476697ed9ada120566076c7996aa68)