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