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