Sr Examen
Integral de (x-0,5*x*x)*sin(pi*m*x/2) dx
Solución
Ha introducido
[src]
2 / | | / x \ /pi*m*x\ | |x - -*x|*sin|------| dx | \ 2 / \ 2 / | / 0
$$\int\limits_{0}^{2} \left(- \frac{x}{2} x + x\right) \sin{\left(\frac{x \pi m}{2} \right)}\, dx$$
Integral((x - x/2*x)*sin(((pi*m)*x)/2), (x, 0, 2))
Respuesta (Indefinida)
[src]
// 0 for m = 0\ // 0 for m = 0\
|| | || |
// 0 for m = 0\ 2 || /pi*m*x\ | || // /pi*m*x\ /pi*m*x\ \ |
|| | x *|<-2*cos|------| | || ||4*cos|------| 2*x*sin|------| | |
/ || // /pi*m*x\ \ | // 0 for m = 0\ || \ 2 / | || || \ 2 / \ 2 / | |
| || ||2*sin|------| | | || | ||-------------- otherwise| || ||------------- + --------------- for m != 0| |
| / x \ /pi*m*x\ || || \ 2 / pi*m | | || /pi*m*x\ | \\ pi*m / || || 2 2 pi*m | |
| |x - -*x|*sin|------| dx = C - |<-2*|<------------- for ---- != 0| | + x*|<-2*cos|------| | - ------------------------------- + |<-2*|< pi *m | |
| \ 2 / \ 2 / || || pi*m 2 | | || \ 2 / | 2 || || | |
| || || | | ||-------------- otherwise| || || 2 | |
/ || \\ x otherwise / | \\ pi*m / || || x | |
||---------------------------------- otherwise| || || -- otherwise | |
\\ pi*m / || \\ 2 / |
||------------------------------------------------- otherwise|
\\ pi*m /
$$\int \left(- \frac{x}{2} x + x\right) \sin{\left(\frac{x \pi m}{2} \right)}\, dx = C - \frac{x^{2} \left(\begin{cases} 0 & \text{for}\: m = 0 \\- \frac{2 \cos{\left(\frac{\pi m x}{2} \right)}}{\pi m} & \text{otherwise} \end{cases}\right)}{2} + x \left(\begin{cases} 0 & \text{for}\: m = 0 \\- \frac{2 \cos{\left(\frac{\pi m x}{2} \right)}}{\pi m} & \text{otherwise} \end{cases}\right) - \begin{cases} 0 & \text{for}\: m = 0 \\- \frac{2 \left(\begin{cases} \frac{2 \sin{\left(\frac{\pi m x}{2} \right)}}{\pi m} & \text{for}\: \frac{\pi m}{2} \neq 0 \\x & \text{otherwise} \end{cases}\right)}{\pi m} & \text{otherwise} \end{cases} + \begin{cases} 0 & \text{for}\: m = 0 \\- \frac{2 \left(\begin{cases} \frac{2 x \sin{\left(\frac{\pi m x}{2} \right)}}{\pi m} + \frac{4 \cos{\left(\frac{\pi m x}{2} \right)}}{\pi^{2} m^{2}} & \text{for}\: m \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases}\right)}{\pi m} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ 8 8*cos(pi*m) 4*sin(pi*m) |------ - ----------- - ----------- for And(m > -oo, m < oo, m != 0) | 3 3 3 3 2 2
$$\begin{cases} - \frac{4 \sin{\left(\pi m \right)}}{\pi^{2} m^{2}} - \frac{8 \cos{\left(\pi m \right)}}{\pi^{3} m^{3}} + \frac{8}{\pi^{3} m^{3}} & \text{for}\: m > -\infty \wedge m < \infty \wedge m \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
=
/ 8 8*cos(pi*m) 4*sin(pi*m) |------ - ----------- - ----------- for And(m > -oo, m < oo, m != 0) | 3 3 3 3 2 2
$$\begin{cases} - \frac{4 \sin{\left(\pi m \right)}}{\pi^{2} m^{2}} - \frac{8 \cos{\left(\pi m \right)}}{\pi^{3} m^{3}} + \frac{8}{\pi^{3} m^{3}} & \text{for}\: m > -\infty \wedge m < \infty \wedge m \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((8/(pi^3*m^3) - 8*cos(pi*m)/(pi^3*m^3) - 4*sin(pi*m)/(pi^2*m^2), (m > -oo)∧(m < oo)∧(Ne(m, 0))), (0, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.