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