Is There a Trick to Simplifying Integrals of Complex Numbers?

[x-is-y]

Suppose we want to find

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

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

$$\int e^x \cos{x} \ dx = \int e^x e^{ix} \ dx = \frac{e^x e^{ix}}{1+i}$$

Doing this gives us
$$(1/2) e^x e^{ix} (1 - i)$$
$$(1/2) e^x (\cos{x} + i\sin{x})(1 - i)$$
$$(1/2) e^x (\cos{x} - i\cos{x} + i\sin{x} + \sin{x})$$

Hence we find that

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

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)

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

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 $$e^x \cos{x}$$ and $$e^x \sin{x}$$ at the same time ... I can't think of any way why this shouldn't be ok.

 

[morphism]

That's a completely valid technique.

 

[MathematicalPhysicist]

just one thing about notation.
it's not correct to equate that the integral of e^x(1+i) eqauls the integral of e^xcos(x), but rather that the real part of the former intergal differs from the latter by a cosntant.
besides this, looks like a nice idea, which can be very useful.