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