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