Sr Examen
Integral de (3*x-3)*sin(2*pi*n*x/3) dx
Solución
Ha introducido
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1 / | | /2*pi*n*x\ | (3*x - 3)*sin|--------| dx | \ 3 / | / 0
$$\int\limits_{0}^{1} \left(3 x - 3\right) \sin{\left(\frac{x 2 \pi n}{3} \right)}\, dx$$
Integral((3*x - 3)*sin((((2*pi)*n)*x)/3), (x, 0, 1))
Respuesta (Indefinida)
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// 0 for n = 0\
|| |
/ || // /2*pi*n*x\ \ | // 0 for n = 0\ // 0 for n = 0\
| || ||3*sin|--------| | | || | || |
| /2*pi*n*x\ || || \ 3 / 2*pi*n | | || /2*pi*n*x\ | || /2*pi*n*x\ |
| (3*x - 3)*sin|--------| dx = C - 3*|<-3*|<--------------- for ------ != 0| | - 3*|<-3*cos|--------| | + 3*x*|<-3*cos|--------| |
| \ 3 / || || 2*pi*n 3 | | || \ 3 / | || \ 3 / |
| || || | | ||---------------- otherwise| ||---------------- otherwise|
/ || \\ x otherwise / | \\ 2*pi*n / \\ 2*pi*n /
||-------------------------------------- otherwise|
\\ 2*pi*n /
$$\int \left(3 x - 3\right) \sin{\left(\frac{x 2 \pi n}{3} \right)}\, dx = C + 3 x \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{3 \cos{\left(\frac{2 \pi n x}{3} \right)}}{2 \pi n} & \text{otherwise} \end{cases}\right) - 3 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{3 \left(\begin{cases} \frac{3 \sin{\left(\frac{2 \pi n x}{3} \right)}}{2 \pi n} & \text{for}\: \frac{2 \pi n}{3} \neq 0 \\x & \text{otherwise} \end{cases}\right)}{2 \pi n} & \text{otherwise} \end{cases}\right) - 3 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{3 \cos{\left(\frac{2 \pi n x}{3} \right)}}{2 \pi n} & \text{otherwise} \end{cases}\right)$$
Respuesta
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/ /2*pi*n\ | 27*sin|------| | 9 \ 3 / |- ------ + -------------- for And(n > -oo, n < oo, n != 0) < 2*pi*n 2 2 | 4*pi *n | | 0 otherwise \
$$\begin{cases} - \frac{9}{2 \pi n} + \frac{27 \sin{\left(\frac{2 \pi n}{3} \right)}}{4 \pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
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/ /2*pi*n\ | 27*sin|------| | 9 \ 3 / |- ------ + -------------- for And(n > -oo, n < oo, n != 0) < 2*pi*n 2 2 | 4*pi *n | | 0 otherwise \
$$\begin{cases} - \frac{9}{2 \pi n} + \frac{27 \sin{\left(\frac{2 \pi n}{3} \right)}}{4 \pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((-9/(2*pi*n) + 27*sin(2*pi*n/3)/(4*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.