Sr Examen
Integral de (2/3)*(3-x)*sin(pi*n*x/3) dx
Solución
Ha introducido
[src]
3 / | | 2*(3 - x) /pi*n*x\ | ---------*sin|------| dx | 3 \ 3 / | / 1
$$\int\limits_{1}^{3} \frac{2 \left(3 - x\right)}{3} \sin{\left(\frac{x \pi n}{3} \right)}\, dx$$
Integral((2*(3 - x)/3)*sin(((pi*n)*x)/3), (x, 1, 3))
Respuesta (Indefinida)
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// 0 for n = 0\
|| |
|| // /pi*n*x\ \ |
|| ||3*sin|------| | |
|| || \ 3 / pi*n | | // 0 for n = 0\
2*|<-3*|<------------- for ---- != 0| | || |
|| || pi*n 3 | | || /pi*n*x\ |
|| || | | 2*x*|<-3*cos|------| |
/ // 0 for n = 0\ || \\ x otherwise / | || \ 3 / |
| || | ||---------------------------------- otherwise| ||-------------- otherwise|
| 2*(3 - x) /pi*n*x\ || /pi*n*x\ | \\ pi*n / \\ pi*n /
| ---------*sin|------| dx = C + 2*|<-3*cos|------| | + -------------------------------------------------- - --------------------------------
| 3 \ 3 / || \ 3 / | 3 3
| ||-------------- otherwise|
/ \\ pi*n /
$$\int \frac{2 \left(3 - x\right)}{3} \sin{\left(\frac{x \pi n}{3} \right)}\, dx = C - \frac{2 x \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{3 \cos{\left(\frac{\pi n x}{3} \right)}}{\pi n} & \text{otherwise} \end{cases}\right)}{3} + \frac{2 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{3 \left(\begin{cases} \frac{3 \sin{\left(\frac{\pi n x}{3} \right)}}{\pi n} & \text{for}\: \frac{\pi n}{3} \neq 0 \\x & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}\right)}{3} + 2 \left(\begin{cases} 0 & \text{for}\: n = 0 \\- \frac{3 \cos{\left(\frac{\pi n x}{3} \right)}}{\pi n} & \text{otherwise} \end{cases}\right)$$
Respuesta
[src]
/ /pi*n\ /pi*n\ | 4*cos|----| 6*sin|----| | 6*sin(pi*n) \ 3 / \ 3 / |- ----------- + ----------- + ----------- for And(n > -oo, n < oo, n != 0) < 2 2 pi*n 2 2 | pi *n pi *n | | 0 otherwise \
$$\begin{cases} \frac{4 \cos{\left(\frac{\pi n}{3} \right)}}{\pi n} + \frac{6 \sin{\left(\frac{\pi n}{3} \right)}}{\pi^{2} n^{2}} - \frac{6 \sin{\left(\pi n \right)}}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
=
/ /pi*n\ /pi*n\ | 4*cos|----| 6*sin|----| | 6*sin(pi*n) \ 3 / \ 3 / |- ----------- + ----------- + ----------- for And(n > -oo, n < oo, n != 0) < 2 2 pi*n 2 2 | pi *n pi *n | | 0 otherwise \
$$\begin{cases} \frac{4 \cos{\left(\frac{\pi n}{3} \right)}}{\pi n} + \frac{6 \sin{\left(\frac{\pi n}{3} \right)}}{\pi^{2} n^{2}} - \frac{6 \sin{\left(\pi n \right)}}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((-6*sin(pi*n)/(pi^2*n^2) + 4*cos(pi*n/3)/(pi*n) + 6*sin(pi*n/3)/(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.