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- Improper integral
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How to use it?
- Integral of d{x}:
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Integral of 1/(2x-1)
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Integral of 1/x(x+1)
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Integral of sen(x)
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Integral of dx/x²
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Identical expressions
- (l/ two)sin(npix/l)dx
- (l divide by 2) sinus of (n Pi x divide by l)dx
- (l divide by two) sinus of (n Pi x divide by l)dx
- l/2sinnpix/ldx
- (l divide by 2)sin(npix divide by l)dx
Integral of (l/2)sin(npix/l)dx dx
The solution
You have entered
[src]
l - 2 / | | l /n*pi*x\ | -*sin|------| dx | 2 \ l / | / 0
$$\int\limits_{0}^{\frac{l}{2}} \frac{l}{2} \sin{\left(\frac{x \pi n}{l} \right)}\, dx$$
Integral((l/2)*sin(((n*pi)*x)/l), (x, 0, l/2))
The answer (Indefinite)
[src]
// 0 for n = 0\
|| |
|| /n*pi*x\ |
l*|<-l*cos|------| |
/ || \ l / |
| ||--------------- otherwise|
| l /n*pi*x\ \\ pi*n /
| -*sin|------| dx = C + -------------------------------
| 2 \ l / 2
|
/
$$\int \frac{l}{2} \sin{\left(\frac{x \pi n}{l} \right)}\, dx = C + \frac{l \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{l \cos{\left(\frac{x \pi n}{l} \right)}}{\pi n} & \text{otherwise} \end{cases}\right)}{2}$$
The answer
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/ 2 /pi*n\ | 2 l *cos|----| | l \ 2 / pi*n <------ - ------------ for ---- != 0 |2*pi*n 2*pi*n l | \ 0 otherwise
$$\begin{cases} - \frac{l^{2} \cos{\left(\frac{\pi n}{2} \right)}}{2 \pi n} + \frac{l^{2}}{2 \pi n} & \text{for}\: \frac{\pi n}{l} \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
=
/ 2 /pi*n\ | 2 l *cos|----| | l \ 2 / pi*n <------ - ------------ for ---- != 0 |2*pi*n 2*pi*n l | \ 0 otherwise
$$\begin{cases} - \frac{l^{2} \cos{\left(\frac{\pi n}{2} \right)}}{2 \pi n} + \frac{l^{2}}{2 \pi n} & \text{for}\: \frac{\pi n}{l} \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((l^2/(2*pi*n) - l^2*cos(pi*n/2)/(2*pi*n), Ne(pi*n/l, 0)), (0, True))
Use the examples entering the upper and lower limits of integration.