Suppose we want to find(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \int e^x \cos{x} \ dx [/tex]

We know from [tex] e^{ix} = \cos{x} + i\sin{x} [/tex] that the real part of [tex] e^{ix} [/tex] equals [tex] \cos{x} [/tex]. So suppose we want to find that integral, is it ok to study the real part of [tex] e^x \cdot e^{ix} [/tex]? In that case we get

[tex] \int e^x \cos{x} \ dx = \int e^x e^{ix} \ dx = \frac{e^x e^{ix}}{1+i}[/tex]

Doing this gives us

[tex] (1/2) e^x e^{ix} (1 - i)[/tex]

[tex] (1/2) e^x (\cos{x} + i\sin{x})(1 - i)[/tex]

[tex] (1/2) e^x (\cos{x} - i\cos{x} + i\sin{x} + \sin{x})[/tex]

Hence we find that

[tex] \int e^x \cos{x} \ dx = (1/2)e^x (\cos{x} + \sin{x}) [/tex]

Which is indeed the result we get from using integration by parts. We also get that the imaginary part is (same as by integration by parts)

[tex] \int e^x \sin{x} \ dx = (1/2) e^x (\sin{x} - \cos{x}) [/tex].

But is this technique ok and is there any more examples of this technique? Is there a name for this? Doing this instead of using integration by parts we get the integrals of [tex] e^x \cos{x} [/tex] and [tex] e^x \sin{x} [/tex] at the same time ... I can't think of any way why this shouldn't be ok.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Complex number in integral

**Physics Forums | Science Articles, Homework Help, Discussion**