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