# Calculus/Integration techniques/Integration by Complexifying

 ← Integration techniques/Integration by Parts Calculus Integration techniques/Partial Fraction Decomposition → Integration techniques/Integration by Complexifying

This technique requires an understanding and recognition of complex numbers. Specifically Euler's formula:

$\cos(\theta )+i\sin(\theta )=e^{\theta i}$ Recognize, for example, that the real portion:

${\text{Re}}\left\{e^{\theta i}\right\}=\cos(\theta )$ Given an integral of the general form:

$\int e^{x}\cos(2x)dx$ We can complexify it:

$\int {\text{Re}}{\Big \{}e^{x}{\big (}\cos(2x)+i\sin(2x){\big )}{\Big \}}dx$ $\int {\text{Re}}{\big \{}e^{x}(e^{2xi}){\big \}}dx$ With basic rules of exponents:

$\int {\text{Re}}\{e^{x+2ix}\}dx$ It can be proven that the "real portion" operator can be moved outside the integral:

${\text{Re}}\left\{\int e^{x(1+2i)}dx\right\}$ The integral easily evaluates:

${\text{Re}}\left\{{\frac {e^{x(1+2i)}}{1+2i}}\right\}$ Multiplying and dividing by $1-2i$ :

${\text{Re}}\left\{{\frac {1-2i}{5}}e^{x(1+2i)}\right\}$ Which can be rewritten as:

${\text{Re}}\left\{{\frac {1-2i}{5}}e^{x}e^{2ix}\right\}$ Applying Euler's forumula:

${\text{Re}}\left\{{\frac {1-2i}{5}}e^{x}{\big (}\cos(2x)+i\sin(2x){\big )}\right\}$ Expanding:

${\text{Re}}\left\{{\frac {e^{x}}{5}}{\big (}\cos(2x)+2\sin(2x){\big )}+i\cdot {\frac {e^{x}}{5}}{\big (}\sin(2x)-2\cos(2x){\big )}\right\}$ Taking the Real part of this expression:

${\frac {e^{x}}{5}}{\big (}\cos(2x)+2\sin(2x){\big )}$ So:

$\int e^{x}\cos(2x)dx={\frac {e^{x}}{5}}{\big (}\cos(2x)+2\sin(2x){\big )}+C$ 