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