Mastering Fourier Integration: Step-by-Step Guide for PDEs | Expert Tips

[nicksauce]

We're doing Fourier series in PDEs, and I am left with needing to integrate:

$$\int_0^{\pi}\cos{nx}\sin{x}dx$$

I completely forget how to integrate this, and I don't have a calculus textbook anymore. Can someone give me a bump in the right direction? I could use maple for the integration, but I would like to know how to integrate it if it came up on an exam.

 

[Hurkyl]

Convert it into exponentials.

 

[nicksauce]

I have not taken a class in complex vars yet, So I am guessing here. Is this correct?

$$\cos{nx}\sin{x} = Im(e^{inx}e^{ix}) - cos(x)sin(nx)$$

 

[rock.freak667]

Well I have not done anything on Fourier integration but when I had to do $$\int sinNxcosMx dx$$

You had to use the fact that $$sin(A+B)+sin(A-B)=2sinAcosB$$

 

[nicksauce]

Well I have not done anything on Fourier integration but when I had to do $$\int sinNxcosMx dx$$

You had to use the fact that $$sin(A+B)+sin(A-B)=2sinAcosB$$

Ah, of course.

 

[Hurkyl]

If you don't already know the exponential form of the trig functions, then practice deriving trig identities on the spot will probably be more fruitful than trying to memorize (or practice deriving) the exponential forms.

But, in case you're curious:

$$\cos x = \frac{e^{ix} + e^{-ix}}{2}$$

$$\sin x = \frac{e^{ix} - e^{-ix}}{2i}$$