Consider $n\in\mathbb{N}$ and $l\in\mathbb{N}$. How do I calculate the integral that follows? $$\int_0^L x\sin\left(\frac{\pi n x}{L}\right)\sin\left(\frac{\pi l x}{L}\right) dx$$
Asked
Active
Viewed 49 times
-1
-
2Try using the product-to-sum trig identity to turn that into a sum of sines – Stephen Donovan May 05 '21 at 21:26
-
Acually, it is a subtraction of cosines. – Angelo May 05 '21 at 21:29
-
$ 2\sin(A) \sin(B) = \cos(A-B) - \cos(A+B)$ – Matha Mota May 05 '21 at 21:30
-
$y=\frac{x\pi}{L}$ makes problem clearer. – herb steinberg May 05 '21 at 21:38
1 Answers
1
Hints:
Note $\cos(ax \pm bx) = \cos ax \cos bx \mp \sin ax \sin bx$, so $$\sin ax \sin bx = \tfrac12\cos(ax - bx)-\tfrac12\cos(ax+bx)$$
Note also that you can integrate by parts: $$\int x\cos kx\; dx = \int x\;d(\tfrac1k\sin kx) = \tfrac1k x\sin kx - \int\tfrac1k \sin kx \; dx = \tfrac1k x\sin kx + \tfrac1{k^2}\cos kx$$
With these, you can perform the integration easily.
MPW
- 43,638