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