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