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the sign was wrong, we get a minus sign upon multiplying with the conjugate |
some cleanup; more is needed |
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Cosine is the real portion of a complex number written in cos x + i sin x form
: <math>\int e^x \cos x
: <math>=\int e^x \mathrm{Re}\left\{ \cos x + i\cdot \sin x \right\} \, dx</math>
: <math>\int e^x \mathrm{Re}\{ e^{ix} \} \, dx</math>
: <math>\mathrm{Re}\left\{ \int e^x e^{ix} \, dx \right}</math>
: <math>\mathrm{Re}\left\{ \int e^{(i+1)x} \, dx \right\}</math>
This calculation continues as:
: <math>=\mathrm{Re}\left\{ \frac{1}{1+i}\ e^{(1+i)x} \right\}</math>
: <math>=\mathrm{Re}\left\{ (\frac{1}{2}\ - i\frac{1}{2}\ )e^x(\cos x +i\sin x) \right\}</math>
=Re 1/2
▲=1/2 exp(x)*cos(x) + 1/2 exp(x)*sin(x)
[[Category:Integral calculus]]
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