I can solve it another way, but am not sure how to use complex numbers to solve it. Thanks for your help
2026-04-25 02:18:23.1777083503
Use complex number to solve this equation $\int e ^{3x} cos x dx$?
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Using Euler Formula,
$e^{ax}\cos bx=$Real of $\{e^{ax}e^{ibx}\}$
$e^{ax}e^{ibx}=e^{x(a+ib)}\implies\int e^{x(a+ib)}dx=\dfrac{e^{x(a+ib)}}{a+ib}+k$
$=\dfrac{(a-ib)e^{x(a+ib)}}{a^2+b^2}+k$
$=\dfrac{(a-ib)e^{ax}(\cos bx+i\sin bx)}{a^2+b^2}+k$