Sr Examen
Integral de (2/pi)(x^2-pix)*sin(nx) dx
Solución
Ha introducido
[src]
pi / | | 2 / 2 \ | --*\x - pi*x/*sin(n*x) dx | pi | / 0
$$\int\limits_{0}^{\pi} \frac{2}{\pi} \left(x^{2} - \pi x\right) \sin{\left(n x \right)}\, dx$$
Integral(((2/pi)*(x^2 - pi*x))*sin(n*x), (x, 0, pi))
Respuesta (Indefinida)
[src]
// 0 for n = 0\
|| |
|| //cos(n*x) x*sin(n*x) \ | / // 0 for n = 0\ \
|| ||-------- + ---------- for n != 0| | | || | |
|| || 2 n | | | || //sin(n*x) \ | // 0 for n = 0\| // 0 for n = 0\
|| || n | | | || ||-------- for n != 0| | || || 2 || |
- 4*|<-|< | | - 2*pi*|- |<-|< n | | + x*|<-cos(n*x) || + 2*x *|<-cos(n*x) |
|| || 2 | | | || || | | ||---------- otherwise|| ||---------- otherwise|
|| || x | | | || \\ x otherwise / | \\ n /| \\ n /
|| || -- otherwise | | | ||------------------------- otherwise| |
/ || \\ 2 / | \ \\ n / /
| ||-------------------------------------- otherwise|
| 2 / 2 \ \\ n /
| --*\x - pi*x/*sin(n*x) dx = C + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------
| pi pi
|
/
$$\int \frac{2}{\pi} \left(x^{2} - \pi x\right) \sin{\left(n x \right)}\, dx = C + \frac{2 x^{2} \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) - 2 \pi \left(x \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\cos{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) - \begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{\sin{\left(n x \right)}}{n} & \text{for}\: n \neq 0 \\x & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right) - 4 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{\begin{cases} \frac{x \sin{\left(n x \right)}}{n} + \frac{\cos{\left(n x \right)}}{n^{2}} & \text{for}\: n \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right)}{\pi}$$
Respuesta
[src]
/ /2*cos(pi*n) pi*sin(pi*n)\ | 2*|----------- + ------------| | | 3 2 | | 4 \ n n / <- ----- + ------------------------------ for And(n > -oo, n < oo, n != 0) | 3 pi | pi*n | \ 0 otherwise
$$\begin{cases} \frac{2 \left(\frac{\pi \sin{\left(\pi n \right)}}{n^{2}} + \frac{2 \cos{\left(\pi n \right)}}{n^{3}}\right)}{\pi} - \frac{4}{\pi n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
=
/ /2*cos(pi*n) pi*sin(pi*n)\ | 2*|----------- + ------------| | | 3 2 | | 4 \ n n / <- ----- + ------------------------------ for And(n > -oo, n < oo, n != 0) | 3 pi | pi*n | \ 0 otherwise
$$\begin{cases} \frac{2 \left(\frac{\pi \sin{\left(\pi n \right)}}{n^{2}} + \frac{2 \cos{\left(\pi n \right)}}{n^{3}}\right)}{\pi} - \frac{4}{\pi n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((-4/(pi*n^3) + 2*(2*cos(pi*n)/n^3 + pi*sin(pi*n)/n^2)/pi, (n > -oo)∧(n < oo)∧(Ne(n, 0))), (0, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.