Mister Exam

# Integral of cos(npix/l) dx

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### The solution

You have entered [src]
  1
/
|
|     /n*pi*x\
|  cos|------| dx
|     \  l   /
|
/
0                 
$$\int\limits_{0}^{1} \cos{\left(\frac{\pi n x}{l} \right)}\, dx$$
The answer (Indefinite) [src]
  /                     //     /pi*n*x\            \
|                      ||l*sin|------|            |
|    /n*pi*x\          ||     \  l   /            |
| cos|------| dx = C + |<-------------  for n != 0|
|    \  l   /          ||     pi*n                |
|                      ||                         |
/                       \\      x        otherwise /
$${{l\,\sin \left({{n\,\pi\,x}\over{l}}\right)}\over{n\,\pi}}$$
/     /pi*n\
|l*sin|----|
|     \ l  /      pi*n
<-----------  for ---- != 0
|    pi*n          l
|
\     1         otherwise  
$${{l\,\sin \left({{n\,\pi}\over{l}}\right)}\over{n\,\pi}}$$
=
=
/     /pi*n\
|l*sin|----|
|     \ l  /      pi*n
<-----------  for ---- != 0
|    pi*n          l
|
\     1         otherwise  
$$\begin{cases} \frac{l \sin{\left(\frac{\pi n}{l} \right)}}{\pi n} & \text{for}\: \frac{\pi n}{l} \neq 0 \\1 & \text{otherwise} \end{cases}$$

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