Lecture 6 example

[Thomas Nutz]

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!

[Victor Ivrii]

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.