Integration using Euler's formula

Functions containing sine or cosine can be expressed as complex exponentials using Euler's formula.

Example: suppose we wanted to integrate:

${\displaystyle \int e^{x}\cos x\,dx}$

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

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

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

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.

${\displaystyle \int e^{x}\cos x\,dx}$

${\displaystyle =\int e^{x}\mathrm {Re} \left\{\cos x+i\cdot \sin x\right\}\,dx}$

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

${\displaystyle \mathrm {Re} \left\{\int e^{x}e^{ix}\,dx\right\}}$

${\displaystyle \mathrm {Re} \left\{\int e^{(i+1)x}\,dx\right\}}$

This calculation continues as:

${\displaystyle =\mathrm {Re} \left\{{\frac {1}{1+i}}\ e^{(1+i)x}\right\}}$

${\displaystyle =\mathrm {Re} \left\{({\frac {1}{2}}\ -i{\frac {1}{2}}\ )e^{x}(\cos x+i\sin x)\right\}}$

=Re 1/2 exp(x) cos(x)+1/2 i exp(x) sin(x)-1/2*i exp(x) cos(x)+1/2 exp(x) sin(x)

=1/2 exp(x) cos(x) + 1/2 exp(x) sin(x)