Integration Techniques - Trigonometric Function Substitutions - cis(x)

1. Aug 21, 2011

GreenPrint

Hi,

I was wondering, just out of curiosity, what family of integrands can be evaluated by making a cis(x) substitution, cis(x)=cos(x)+isin(x), I can't seem to find any examples or anything at all and was wondering if someone could provide an example or inform of how to make such substitutions.

2. Aug 21, 2011

Gib Z

There isn't really a protypical situation to use $$u = \cos x + i \sin x$$ like there are for other substitution, eg trying $t=\tan x$ if one sees $$\sqrt{1+t^2}$$ in the integrand. Thing will become for clear when you learn that this mysterious function cis(x) is actually just $$e^{ix}$$ and all the rules you've learned about exponentials before will apply again.

3. Aug 21, 2011

GreenPrint

Ya I understand and am familiar with the function. The only problem is that I can't think of a situation period in which I could make such a substitution, nor can I find any examples of when I could do so.

4. Aug 21, 2011

Screwdriver

Actually, I remember reading something relevant to this in J. Nearing's Mathematical Tools for Physics. Consider:

$$\int cos(ax)e^{bx}dx + i \int sin(ax)e^{bx}dx$$

$$\int e^{bx}[cos(ax) + i \, sin(ax)]dx$$

Here we make the substitution $cos(ax) + i \, sin(ax) = cis(ax) = e^{i(ax)}$

$$\int e^{bx} e^{iax} dx$$
$$\int e^{(b + ia)x} dx$$

Now the integral is extremely easy. After evaluation, one can consider the real and imaginary parts to obtain $\int cos(ax)e^{bx}dx$ and $\int sin(ax)e^{bx}dx$ without integrating by parts even once!

Try finding that in a Biology textbook.

5. Aug 21, 2011

GreenPrint

Hmm... Very interesting. Thanks. That's a great integration technique.

6. Aug 21, 2011

Screwdriver

Don't spread it around too much