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