Sr Examen
Integral de 5*pi*x*cos((5*pi)/8)*sin((m*x)/7) dx
Solución
Ha introducido
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7 / | | /5*pi\ /m*x\ | 5*pi*x*cos|----|*sin|---| dx | \ 8 / \ 7 / | / 0
$$\int\limits_{0}^{7} 5 \pi x \cos{\left(\frac{5 \pi}{8} \right)} \sin{\left(\frac{m x}{7} \right)}\, dx$$
Integral((((5*pi)*x)*cos((5*pi)/8))*sin((m*x)/7), (x, 0, 7))
Respuesta (Indefinida)
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/ // 0 for m = 0\ \
| || | |
| || // /m*x\ \ | // 0 for m = 0\|
___________ | || ||7*sin|---| | | || ||
/ ___ | || || \ 7 / m | | || /m*x\ ||
5*pi*\/ 2 - \/ 2 *|- |<-7*|<---------- for - != 0| | + x*|<-7*cos|---| ||
| || || m 7 | | || \ 7 / ||
| || || | | ||----------- otherwise||
/ | || \\ x otherwise / | \\ m /|
| | ||---------------------------- otherwise| |
| /5*pi\ /m*x\ \ \\ m / /
| 5*pi*x*cos|----|*sin|---| dx = C - ------------------------------------------------------------------------------------------------
| \ 8 / \ 7 / 2
|
/
$$\int 5 \pi x \cos{\left(\frac{5 \pi}{8} \right)} \sin{\left(\frac{m x}{7} \right)}\, dx = C - \frac{5 \pi \sqrt{2 - \sqrt{2}} \left(x \left(\begin{cases} 0 & \text{for}\: m = 0 \\- \frac{7 \cos{\left(\frac{m x}{7} \right)}}{m} & \text{otherwise} \end{cases}\right) - \begin{cases} 0 & \text{for}\: m = 0 \\- \frac{7 \left(\begin{cases} \frac{7 \sin{\left(\frac{m x}{7} \right)}}{m} & \text{for}\: \frac{m}{7} \neq 0 \\x & \text{otherwise} \end{cases}\right)}{m} & \text{otherwise} \end{cases}\right)}{2}$$
Respuesta
[src]
/ ___________ | / ___ | / 1 \/ 2 / 49*cos(m) 49*sin(m)\ |-5*pi* / - - ----- *|- --------- + ---------| for And(m > -oo, m < oo, m != 0) < \/ 2 4 | m 2 | | \ m / | | 0 otherwise \
$$\begin{cases} - 5 \pi \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \left(- \frac{49 \cos{\left(m \right)}}{m} + \frac{49 \sin{\left(m \right)}}{m^{2}}\right) & \text{for}\: m > -\infty \wedge m < \infty \wedge m \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
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/ ___________ | / ___ | / 1 \/ 2 / 49*cos(m) 49*sin(m)\ |-5*pi* / - - ----- *|- --------- + ---------| for And(m > -oo, m < oo, m != 0) < \/ 2 4 | m 2 | | \ m / | | 0 otherwise \
$$\begin{cases} - 5 \pi \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \left(- \frac{49 \cos{\left(m \right)}}{m} + \frac{49 \sin{\left(m \right)}}{m^{2}}\right) & \text{for}\: m > -\infty \wedge m < \infty \wedge m \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((-5*pi*sqrt(1/2 - sqrt(2)/4)*(-49*cos(m)/m + 49*sin(m)/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.