Sr Examen
Integral de x*sin((pi*x)/(2l))*sin((pi*x*n)/l) dx
Solución
Ha introducido
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l / | | /pi*x\ /pi*x*n\ | x*sin|----|*sin|------| dx | \2*l / \ l / | / 0
$$\int\limits_{0}^{l} x \sin{\left(\frac{\pi x}{2 l} \right)} \sin{\left(\frac{n \pi x}{l} \right)}\, dx$$
Integral((x*sin((pi*x)/((2*l))))*sin(((pi*x)*n)/l), (x, 0, l))
Respuesta
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/ 2 2 | l l | - -- - --- for n = -1/2 | 4 2 | pi | | 2 2 | l l < -- + --- for n = 1/2 | 4 2 | pi | | 2 2 2 2 2 3 2 | 16*n*l 4*l *sin(pi*n) 16*l *n *sin(pi*n) 16*pi*l *n *cos(pi*n) 4*pi*n*l *cos(pi*n) |- -------------------------- + -------------------------- + -------------------------- - -------------------------- + -------------------------- otherwise | 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 \ pi - 8*pi *n + 16*pi *n pi - 8*pi *n + 16*pi *n pi - 8*pi *n + 16*pi *n pi - 8*pi *n + 16*pi *n pi - 8*pi *n + 16*pi *n
$$\begin{cases} - \frac{l^{2}}{4} - \frac{l^{2}}{\pi^{2}} & \text{for}\: n = - \frac{1}{2} \\\frac{l^{2}}{\pi^{2}} + \frac{l^{2}}{4} & \text{for}\: n = \frac{1}{2} \\- \frac{16 \pi l^{2} n^{3} \cos{\left(\pi n \right)}}{16 \pi^{2} n^{4} - 8 \pi^{2} n^{2} + \pi^{2}} + \frac{16 l^{2} n^{2} \sin{\left(\pi n \right)}}{16 \pi^{2} n^{4} - 8 \pi^{2} n^{2} + \pi^{2}} + \frac{4 \pi l^{2} n \cos{\left(\pi n \right)}}{16 \pi^{2} n^{4} - 8 \pi^{2} n^{2} + \pi^{2}} - \frac{16 l^{2} n}{16 \pi^{2} n^{4} - 8 \pi^{2} n^{2} + \pi^{2}} + \frac{4 l^{2} \sin{\left(\pi n \right)}}{16 \pi^{2} n^{4} - 8 \pi^{2} n^{2} + \pi^{2}} & \text{otherwise} \end{cases}$$
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/ 2 2 | l l | - -- - --- for n = -1/2 | 4 2 | pi | | 2 2 | l l < -- + --- for n = 1/2 | 4 2 | pi | | 2 2 2 2 2 3 2 | 16*n*l 4*l *sin(pi*n) 16*l *n *sin(pi*n) 16*pi*l *n *cos(pi*n) 4*pi*n*l *cos(pi*n) |- -------------------------- + -------------------------- + -------------------------- - -------------------------- + -------------------------- otherwise | 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 2 2 2 2 4 \ pi - 8*pi *n + 16*pi *n pi - 8*pi *n + 16*pi *n pi - 8*pi *n + 16*pi *n pi - 8*pi *n + 16*pi *n pi - 8*pi *n + 16*pi *n
$$\begin{cases} - \frac{l^{2}}{4} - \frac{l^{2}}{\pi^{2}} & \text{for}\: n = - \frac{1}{2} \\\frac{l^{2}}{\pi^{2}} + \frac{l^{2}}{4} & \text{for}\: n = \frac{1}{2} \\- \frac{16 \pi l^{2} n^{3} \cos{\left(\pi n \right)}}{16 \pi^{2} n^{4} - 8 \pi^{2} n^{2} + \pi^{2}} + \frac{16 l^{2} n^{2} \sin{\left(\pi n \right)}}{16 \pi^{2} n^{4} - 8 \pi^{2} n^{2} + \pi^{2}} + \frac{4 \pi l^{2} n \cos{\left(\pi n \right)}}{16 \pi^{2} n^{4} - 8 \pi^{2} n^{2} + \pi^{2}} - \frac{16 l^{2} n}{16 \pi^{2} n^{4} - 8 \pi^{2} n^{2} + \pi^{2}} + \frac{4 l^{2} \sin{\left(\pi n \right)}}{16 \pi^{2} n^{4} - 8 \pi^{2} n^{2} + \pi^{2}} & \text{otherwise} \end{cases}$$
Piecewise((-l^2/4 - l^2/pi^2, n = -1/2), (l^2/4 + l^2/pi^2, n = 1/2), (-16*n*l^2/(pi^2 - 8*pi^2*n^2 + 16*pi^2*n^4) + 4*l^2*sin(pi*n)/(pi^2 - 8*pi^2*n^2 + 16*pi^2*n^4) + 16*l^2*n^2*sin(pi*n)/(pi^2 - 8*pi^2*n^2 + 16*pi^2*n^4) - 16*pi*l^2*n^3*cos(pi*n)/(pi^2 - 8*pi^2*n^2 + 16*pi^2*n^4) + 4*pi*n*l^2*cos(pi*n)/(pi^2 - 8*pi^2*n^2 + 16*pi^2*n^4), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.