Sr Examen
Integral de sin(pi*x)*sin(pi*n*x) dx
Solución
Ha introducido
[src]
1 / | | sin(pi*x)*sin(pi*n*x) dx | / 0
$$\int\limits_{0}^{1} \sin{\left(\pi x \right)} \sin{\left(x \pi n \right)}\, dx$$
Integral(sin(pi*x)*sin((pi*n)*x), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ / pi*x pi*n*x\ / /pi*x pi*n*x\
| 2*tan|- ---- + ------| | 2*tan|---- + ------|
| \ 2 2 / | \ 2 2 /
|------------------------------------------------------------------ for n != 1 |------------------------------------------------------------- for n != -1
< 2/ pi*x pi*n*x\ 2/ pi*x pi*n*x\ < 2/pi*x pi*n*x\ 2/pi*x pi*n*x\
|-pi + pi*n - pi*tan |- ---- + ------| + pi*n*tan |- ---- + ------| |pi + pi*n + pi*tan |---- + ------| + pi*n*tan |---- + ------|
| \ 2 2 / \ 2 2 / | \ 2 2 / \ 2 2 /
/ | |
| \ x otherwise \ x otherwise
| sin(pi*x)*sin(pi*n*x) dx = C + ------------------------------------------------------------------------------- - ---------------------------------------------------------------------------
| 2 2
/
$$\int \sin{\left(\pi x \right)} \sin{\left(x \pi n \right)}\, dx = C + \frac{\begin{cases} \frac{2 \tan{\left(\frac{\pi n x}{2} - \frac{\pi x}{2} \right)}}{\pi n \tan^{2}{\left(\frac{\pi n x}{2} - \frac{\pi x}{2} \right)} + \pi n - \pi \tan^{2}{\left(\frac{\pi n x}{2} - \frac{\pi x}{2} \right)} - \pi} & \text{for}\: n \neq 1 \\x & \text{otherwise} \end{cases}}{2} - \frac{\begin{cases} \frac{2 \tan{\left(\frac{\pi n x}{2} + \frac{\pi x}{2} \right)}}{\pi n \tan^{2}{\left(\frac{\pi n x}{2} + \frac{\pi x}{2} \right)} + \pi n + \pi \tan^{2}{\left(\frac{\pi n x}{2} + \frac{\pi x}{2} \right)} + \pi} & \text{for}\: n \neq -1 \\x & \text{otherwise} \end{cases}}{2}$$
Respuesta
[src]
/ -1/2 for n = -1 | | 1/2 for n = 1 | <-sin(pi*n) |----------- otherwise | 2 |-pi + pi*n \
$$\begin{cases} - \frac{1}{2} & \text{for}\: n = -1 \\\frac{1}{2} & \text{for}\: n = 1 \\- \frac{\sin{\left(\pi n \right)}}{\pi n^{2} - \pi} & \text{otherwise} \end{cases}$$
=
=
/ -1/2 for n = -1 | | 1/2 for n = 1 | <-sin(pi*n) |----------- otherwise | 2 |-pi + pi*n \
$$\begin{cases} - \frac{1}{2} & \text{for}\: n = -1 \\\frac{1}{2} & \text{for}\: n = 1 \\- \frac{\sin{\left(\pi n \right)}}{\pi n^{2} - \pi} & \text{otherwise} \end{cases}$$
Piecewise((-1/2, n = -1), (1/2, n = 1), (-sin(pi*n)/(-pi + pi*n^2), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.