Sr Examen
Integral de l^2*x^2*cos(pi*k*x/l) dx
Solución
Ha introducido
[src]
l / | | 2 2 /pi*k*x\ | l *x *cos|------| dx | \ l / | / 0
$$\int\limits_{0}^{l} l^{2} x^{2} \cos{\left(\frac{x \pi k}{l} \right)}\, dx$$
Integral((l^2*x^2)*cos(((pi*k)*x)/l), (x, 0, l))
Respuesta (Indefinida)
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// 3 \
|| x |
|| -- for k = 0|
|| 3 |
|| |
/ || // 2 /pi*k*x\ /pi*k*x\ \ | // x for k = 0\
| || ||l *sin|------| l*x*cos|------| | | || |
| 2 2 /pi*k*x\ 2 || || \ l / \ l / | | 2 2 || /pi*k*x\ |
| l *x *cos|------| dx = C - 2*l *|< ||-------------- - --------------- for k != 0| | + l *x *|
$$\int l^{2} x^{2} \cos{\left(\frac{x \pi k}{l} \right)}\, dx = C + l^{2} x^{2} \left(\begin{cases} x & \text{for}\: k = 0 \\\frac{l \sin{\left(\frac{\pi k x}{l} \right)}}{\pi k} & \text{otherwise} \end{cases}\right) - 2 l^{2} \left(\begin{cases} \frac{x^{3}}{3} & \text{for}\: k = 0 \\\frac{l \left(\begin{cases} - \frac{l x \cos{\left(\frac{\pi k x}{l} \right)}}{\pi k} + \frac{l^{2} \sin{\left(\frac{\pi k x}{l} \right)}}{\pi^{2} k^{2}} & \text{for}\: k \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{\pi k} & \text{otherwise} \end{cases}\right)$$
Respuesta
[src]
/ / 3 3 3 \ | 2 |l *sin(pi*k) 2*l *sin(pi*k) 2*l *cos(pi*k)| |l *|------------ - -------------- + --------------| for And(k > -oo, k < oo, k != 0) | | pi*k 3 3 2 2 | | \ pi *k pi *k / < | 5 | l | -- otherwise | 3 \
$$\begin{cases} l^{2} \left(\frac{l^{3} \sin{\left(\pi k \right)}}{\pi k} + \frac{2 l^{3} \cos{\left(\pi k \right)}}{\pi^{2} k^{2}} - \frac{2 l^{3} \sin{\left(\pi k \right)}}{\pi^{3} k^{3}}\right) & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\\frac{l^{5}}{3} & \text{otherwise} \end{cases}$$
=
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/ / 3 3 3 \ | 2 |l *sin(pi*k) 2*l *sin(pi*k) 2*l *cos(pi*k)| |l *|------------ - -------------- + --------------| for And(k > -oo, k < oo, k != 0) | | pi*k 3 3 2 2 | | \ pi *k pi *k / < | 5 | l | -- otherwise | 3 \
$$\begin{cases} l^{2} \left(\frac{l^{3} \sin{\left(\pi k \right)}}{\pi k} + \frac{2 l^{3} \cos{\left(\pi k \right)}}{\pi^{2} k^{2}} - \frac{2 l^{3} \sin{\left(\pi k \right)}}{\pi^{3} k^{3}}\right) & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\\frac{l^{5}}{3} & \text{otherwise} \end{cases}$$
Piecewise((l^2*(l^3*sin(pi*k)/(pi*k) - 2*l^3*sin(pi*k)/(pi^3*k^3) + 2*l^3*cos(pi*k)/(pi^2*k^2)), (k > -oo)∧(k < oo)∧(Ne(k, 0))), (l^5/3, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.