Sr Examen
Integral de xcosnx dx
Solución
Ha introducido
[src]
2*pi / | | x*cos(n*x) dx | / 0
$$\int\limits_{0}^{2 \pi} x \cos{\left(n x \right)}\, dx$$
Integral(x*cos(n*x), (x, 0, 2*pi))
Respuesta (Indefinida)
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// 2 \
|| x |
|| -- for n = 0|
|| 2 |
/ || | // x for n = 0\
| ||/-cos(n*x) | || |
| x*cos(n*x) dx = C - |<|---------- for n != 0 | + x*|
$$\int x \cos{\left(n x \right)}\, dx = C + x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{\cos{\left(n x \right)}}{n} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ 1 cos(2*pi*n) 2*pi*sin(2*pi*n) |- -- + ----------- + ---------------- for And(n > -oo, n < oo, n != 0) | 2 2 n < n n | | 2 \ 2*pi otherwise
$$\begin{cases} \frac{2 \pi \sin{\left(2 \pi n \right)}}{n} + \frac{\cos{\left(2 \pi n \right)}}{n^{2}} - \frac{1}{n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\2 \pi^{2} & \text{otherwise} \end{cases}$$
=
=
/ 1 cos(2*pi*n) 2*pi*sin(2*pi*n) |- -- + ----------- + ---------------- for And(n > -oo, n < oo, n != 0) | 2 2 n < n n | | 2 \ 2*pi otherwise
$$\begin{cases} \frac{2 \pi \sin{\left(2 \pi n \right)}}{n} + \frac{\cos{\left(2 \pi n \right)}}{n^{2}} - \frac{1}{n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\2 \pi^{2} & \text{otherwise} \end{cases}$$
Piecewise((-1/n^2 + cos(2*pi*n)/n^2 + 2*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.