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