Sr Examen
Integral de 2/l*x^2*sin((n*pi*x)/l) dx
Solución
Ha introducido
[src]
1 / | | 2 2 /n*pi*x\ | -*x *sin|------| dx | l \ l / | / 0
$$\int\limits_{0}^{1} \frac{2}{l} x^{2} \sin{\left(\frac{x \pi n}{l} \right)}\, dx$$
Integral(((2/l)*x^2)*sin(((n*pi)*x)/l), (x, 0, 1))
Respuesta (Indefinida)
[src]
// 0 for n = 0\
|| |
|| // 2 /pi*n*x\ /pi*n*x\ \ |
|| ||l *cos|------| l*x*sin|------| | |
|| || \ l / \ l / | |
|| ||-------------- + --------------- for n != 0| |
|| || 2 2 pi*n | |
4*|<-l*|< pi *n | |
|| || | | // 0 for n = 0\
|| || 2 | | || |
|| || x | | 2 || /pi*n*x\ |
|| || -- otherwise | | 2*x *|<-l*cos|------| |
/ || \\ 2 / | || \ l / |
| ||--------------------------------------------------- otherwise| ||--------------- otherwise|
| 2 2 /n*pi*x\ \\ pi*n / \\ pi*n /
| -*x *sin|------| dx = C - ------------------------------------------------------------------- + ----------------------------------
| l \ l / l l
|
/
$$\int \frac{2}{l} x^{2} \sin{\left(\frac{x \pi n}{l} \right)}\, dx = C + \frac{2 x^{2} \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{l \cos{\left(\frac{\pi n x}{l} \right)}}{\pi n} & \text{otherwise} \end{cases}\right)}{l} - \frac{4 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{l \left(\begin{cases} \frac{l^{2} \cos{\left(\frac{\pi n x}{l} \right)}}{\pi^{2} n^{2}} + \frac{l x \sin{\left(\frac{\pi n x}{l} \right)}}{\pi n} & \text{for}\: n \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}\right)}{l}$$
Respuesta
[src]
/ / /pi*n\ 3 /pi*n\ 2 /pi*n\\ | | l*cos|----| 2*l *cos|----| 2*l *sin|----|| | | \ l / \ l / \ l /| |2*|- ----------- + -------------- + --------------| | | pi*n 3 3 2 2 | 2 < \ pi *n pi *n / 4*l |--------------------------------------------------- - ------ for And(n > -oo, n < oo, n != 0) | l 3 3 | pi *n | \ 0 otherwise
$$\begin{cases} - \frac{4 l^{2}}{\pi^{3} n^{3}} + \frac{2 \left(\frac{2 l^{3} \cos{\left(\frac{\pi n}{l} \right)}}{\pi^{3} n^{3}} + \frac{2 l^{2} \sin{\left(\frac{\pi n}{l} \right)}}{\pi^{2} n^{2}} - \frac{l \cos{\left(\frac{\pi n}{l} \right)}}{\pi n}\right)}{l} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
=
/ / /pi*n\ 3 /pi*n\ 2 /pi*n\\ | | l*cos|----| 2*l *cos|----| 2*l *sin|----|| | | \ l / \ l / \ l /| |2*|- ----------- + -------------- + --------------| | | pi*n 3 3 2 2 | 2 < \ pi *n pi *n / 4*l |--------------------------------------------------- - ------ for And(n > -oo, n < oo, n != 0) | l 3 3 | pi *n | \ 0 otherwise
$$\begin{cases} - \frac{4 l^{2}}{\pi^{3} n^{3}} + \frac{2 \left(\frac{2 l^{3} \cos{\left(\frac{\pi n}{l} \right)}}{\pi^{3} n^{3}} + \frac{2 l^{2} \sin{\left(\frac{\pi n}{l} \right)}}{\pi^{2} n^{2}} - \frac{l \cos{\left(\frac{\pi n}{l} \right)}}{\pi n}\right)}{l} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((2*(-l*cos(pi*n/l)/(pi*n) + 2*l^3*cos(pi*n/l)/(pi^3*n^3) + 2*l^2*sin(pi*n/l)/(pi^2*n^2))/l - 4*l^2/(pi^3*n^3), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (0, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.