How to solve this integrals (delta Dirac)

Question

i have the following problem :

$$\int_0^L\sum_{i=1}^{n=3} \delta\left(x-x_{i}\right) \cos(2\pi x/L ) \left\{ A_{1}'(t) \cos \left(\frac{2 \pi x}{L}\right) + B_{1}'(t) \sin \left(\frac{2 \pi x}{L}\right)\right\} dx $$

With $x_i={L/6,3L/6,5L/6}$.

basically, I did this:

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$$= \int_0^L\sum_{i=1}^{n=3} \delta\left(x-x_{i}\right)\left\{ A_{1}'(t) \cos ^2\left(\frac{2 \pi x}{L}\right)+B_{1}'(t) \sin \left(\frac{2 \pi x}{L}\right)\cos \left(\frac{2 \pi x}{L}\right)\right\}dx $$

$$=\sum_{i=1}^{n=3}\left\{ A_{1}'(t) \cos ^2\left(\frac{2 \pi x_i}{L}\right)+B_{1}'(t) \sin \left(\frac{2 \pi x}{L}\right)\cos \left(\frac{2 \pi x_i}{L}\right)\right\} $$

$$= \frac{3}{2}A_1'(t)+\frac{3}{2}B_1'(t).$$

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But this is far from what my teacher found, he basically found $3A_1'(t)$, anyone can help me or explain to me my mistake(s)? thanks!

Answer 1

It is $\frac{3}{2}A_1'(t)$.

\begin{align*} &\int_0^L \sum_{i=1}^3 \delta(x-x_i) \cos \left( \frac{2\pi x}{L} \right) \left( A_1'(t) \cos \left( \frac{2\pi x}{L} \right) + B_1'(t) \sin \left( \frac{2\pi x}{L} \right)\right) \,\mathrm{d}x \\ &= \sum_{i=1}^3 \int_0^L \delta(x-x_i) \cos \left( \frac{2\pi x}{L} \right) \left( A_1'(t) \cos \left( \frac{2\pi x}{L} \right) + B_1'(t) \sin \left( \frac{2\pi x}{L} \right)\right) \,\mathrm{d}x \\ &= \sum_{i=1}^3 \cos \left( \frac{2\pi x_i}{L} \right) \left( A_1'(t) \cos \left( \frac{2\pi x_i}{L} \right) + B_1'(t) \sin \left( \frac{2\pi x_i}{L} \right)\right) \Theta(L-x_i) \\ &= \cos \left( \frac{2\pi x_1}{L} \right) \left( A_1'(t) \cos \left( \frac{2\pi x_1}{L} \right) + B_1'(t) \sin \left( \frac{2\pi x_1}{L} \right)\right) \Theta(L-x_1) \\ & \qquad + \cos \left( \frac{2\pi x_2}{L} \right) \left( A_1'(t) \cos \left( \frac{2\pi x_2}{L} \right) + B_1'(t) \sin \left( \frac{2\pi x_2}{L} \right)\right) \Theta(L-x_2) \\ & \qquad + \cos \left( \frac{2\pi x_3}{L} \right) \left( A_1'(t) \cos \left( \frac{2\pi x_3}{L} \right) + B_1'(t) \sin \left( \frac{2\pi x_3}{L} \right)\right) \Theta(L-x_3) \\ &= \cos \left( \frac{2\pi \frac{L}{6}}{L} \right) \left( A_1'(t) \cos \left( \frac{2\pi \frac{L}{6}}{L} \right) + B_1'(t) \sin \left( \frac{2\pi \frac{L}{6}}{L} \right)\right) \Theta(L-\frac{L}{6}) \\ & \qquad + \cos \left( \frac{2\pi \frac{3L}{6}}{L} \right) \left( A_1'(t) \cos \left( \frac{2\pi \frac{3L}{6}}{L} \right) + B_1'(t) \sin \left( \frac{2\pi \frac{3L}{6}}{L} \right)\right) \Theta(L-\frac{3L}{6}) \\ & \qquad + \cos \left( \frac{2\pi \frac{5L}{6}}{L} \right) \left( A_1'(t) \cos \left( \frac{2\pi \frac{5L}{6}}{L} \right) + B_1'(t) \sin \left( \frac{2\pi \frac{5L}{6}}{L} \right)\right) \Theta(L-\frac{5L}{6}) \\ &= \cos \left( \frac{\pi}{3} \right) \left( A_1'(t) \cos \left( \frac{\pi}{3} \right) + B_1'(t) \sin \left( \frac{\pi}{3} \right)\right) \cdot 1 \\ & \qquad + \cos \left( \pi \right) \left( A_1'(t) \cos \left( \pi \right) + B_1'(t) \sin \left( \pi \right)\right) \cdot 1 \\ & \qquad + \cos \left( \frac{5\pi}{3} \right) \left( A_1'(t) \cos \left( \frac{5\pi}{3} \right) + B_1'(t) \sin \left( {5\pi}{3} \right)\right) \cdot 1 \\ &= \frac{1}{2} \left( \frac{1}{2} A_1'(t) + \frac{\sqrt{3}}{2} B_1'(t) \right) \\ & \qquad + (-1) \left( -A_1'(t) + 0 B_1'(t) \right) \\ & \qquad + \frac{1}{2} \left( \frac{1}{2} A_1'(t) + \frac{-\sqrt{3}}{2} B_1'(t) \right) \\ &= \frac{1}{4} A_1'(t) + \frac{\sqrt{3}}{4} B_1'(t) + A_1'(t) + \frac{1}{4} A_1'(t) + \frac{-\sqrt{3}}{4} B_1'(t) \\ &= \frac{3}{2} A_1'(t) \text{.} \end{align*}