Mister Exam
Integral of sin(npix/l) dx
The solution
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1 / | | /n*pi*x\ | sin|------| dx | \ l / | / 0
$$\int\limits_{0}^{1} \sin{\left(\frac{\pi n x}{l} \right)}\, dx$$
The answer (Indefinite)
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/ // /pi*n*x\ \ | ||-l*cos|------| | | /n*pi*x\ || \ l / | | sin|------| dx = C + |<--------------- for n != 0| | \ l / || pi*n | | || | / \\ 0 otherwise /
$$-{{l\,\cos \left({{n\,\pi\,x}\over{l}}\right)}\over{n\,\pi}}$$
The answer
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/ /pi*n\ | l*cos|----| | l \ l / pi*n <---- - ----------- for ---- != 0 |pi*n pi*n l | \ 0 otherwise
$${{l}\over{n\,\pi}}-{{l\,\cos \left({{n\,\pi}\over{l}}\right)}\over{
n\,\pi}}$$
=
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/ /pi*n\ | l*cos|----| | l \ l / pi*n <---- - ----------- for ---- != 0 |pi*n pi*n l | \ 0 otherwise
$$\begin{cases} - \frac{l \cos{\left(\frac{\pi n}{l} \right)}}{\pi n} + \frac{l}{\pi n} & \text{for}\: \frac{\pi n}{l} \neq 0 \\0 & \text{otherwise} \end{cases}$$
Use the examples entering the upper and lower limits of integration.