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