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