Integrating trigonometric products as complex exponentials

Functions containing sine or cosine can be expressed as complex exponentials using
Euler's formula. Example: suppose we wanted to integrate:

: $int\; e^x\; cos\; x\; ,\; dx$

Then the cosine function can be expressed in its Euler form: $cos\; heta\; =\; frac\{e^\{i\; heta\}\; +\; e^\{-i\; heta\{2\}$

: $int\; e^x\; cdot\; frac\{e^\{ix\}\; +\; e^\{-ix\{2\}\; ,\; dx$

: $\{1over\; 2\}\; int\; e^\{x(1+i)\}\; +\; e^\{x(1-i)\}\; ,\; dx$

This is far easier to integrate.

Alternatively, we may also take note of real and imaginary portions of complex numbers

Cosine is the real portion of a complex number written in cos x + i sin x form

$int\; e^x\; cos\; x\; dx\; =$

$int\; e^x\; mathrm\{Re\}\{\; cos\; x\; +\; icdot\; sin\; x\; \}\; dx$

$int\; e^x\; mathrm\{Re\}\{\; e^\{ix\}\; \}\; dx$

$mathrm\{Re\}\{\; int\; e^x\; e^\{ix\}\; dx\; \}$

$mathrm\{Re\}\{\; int\; e^\{(i+1)x\}\; dx\; \}$