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