Mister exam

# Integral sin(npix/l) dx

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### The solution

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  1
/
|
|     /n*pi*x\
|  sin|------| dx
|     \  l   /
|
/
0                 
$$\int\limits_{0}^{1} \sin{\left(\frac{x \pi n}{l} \right)}\, dx$$
Integral(sin(((n*pi)*x)/l), (x, 0, 1))
/            /pi*n\
|       l*cos|----|
| l          \ l  /      pi*n
<---- - -----------  for ---- != 0
|pi*n       pi*n          l
|
\        0             otherwise  
$$\begin{cases} - \frac{l \cos{\left(\frac{\pi n}{l} \right)}}{\pi n} + \frac{l}{\pi n} & \text{for}\: \frac{\pi n}{l} \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
=
/            /pi*n\
|       l*cos|----|
| l          \ l  /      pi*n
<---- - -----------  for ---- != 0
|pi*n       pi*n          l
|
\        0             otherwise  
$$\begin{cases} - \frac{l \cos{\left(\frac{\pi n}{l} \right)}}{\pi n} + \frac{l}{\pi n} & \text{for}\: \frac{\pi n}{l} \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((l/(pi*n) - l*cos(pi*n/l)/(pi*n), Ne(pi*n/l, 0)), (0, True))
  /                     //      /n*pi*x\             \
|                      ||-l*cos|------|             |
|    /n*pi*x\          ||      \  l   /             |
| sin|------| dx = C + |<---------------  for n != 0|
|    \  l   /          ||      pi*n                 |
|                      ||                           |
/                       \\       0         otherwise /
$$\int \sin{\left(\frac{x \pi n}{l} \right)}\, dx = C + \begin{cases} - \frac{l \cos{\left(\frac{x \pi n}{l} \right)}}{\pi n} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}$$

Use the examples entering the upper and lower limits of integration.

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