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