# Toronto Math Forum

## APM346-2012 => APM346 Math => Misc Math => Topic started by: Thomas Nutz on September 27, 2012, 04:23:33 PM

Title: Lecture 6 example
Post by: Thomas Nutz on September 27, 2012, 04:23:33 PM
Hello,
I was going through the 6. lecture notes, where in the end an example is brought up that leads to an integral
$$\frac{1}{4}\int_0^tcos(t')(cos(x-ct+ct')-cos(x+ct-ct'))dt'$$
I was trying to do that integral, but the only way that I could do it was to write out the cosines as complex exponentials, which lead me to eight terms in the end... Is there a cleverer way to do this integral?
Thanks!
Title: Re: Lecture 6 example
Post by: Victor Ivrii on September 27, 2012, 04:49:15 PM
Hello,
I was going through the 6. lecture notes, where in the end an example is brought up that leads to an integral
$$I=\frac{1}{4}\int_0^t\cos(t')\bigl(\cos(x-ct+ct')-\cos(x+ct-ct')\bigr)dt'$$
I was trying to do that integral, but the only way that I could do it was to write out the cosines as complex exponentials, which lead me to eight terms in the end... Is there a cleverer way to do this integral?
Thanks!

In this example $c=2$ helps a bit but you could sea; without it.

The 1-st line -> 2nd (just integration) -> 3rd (formula $\cos (\alpha)-\cos (\beta)= 2\sin \bigl((\beta+\alpha)/2\bigr)\cdot \sin \bigl((\beta-\alpha)/2\bigr)$ and we ge the 4-th line.
Then formula $2\sin( \alpha) \cdot \cos(\beta)=\sin (\alpha+\beta)+\sin (\alpha-\beta)$ and integration.