Integral de x*cos(x)*sin(nx) dx
Solución
Respuesta (Indefinida)
[src]
// 2 2 \
|| x*sin (x) cos(x)*sin(x) x*cos (x) |
|| - --------- - ------------- + --------- for n = -1|
|| 4 4 4 |
|| |
/ || 2 2 |
| || x*cos (x) x*sin (x) cos(x)*sin(x) |
| x*cos(x)*sin(n*x) dx = C + |< - --------- + --------- + ------------- for n = 1 |
| || 4 4 4 |
/ || |
|| 2 2 3 |
||cos(x)*sin(n*x) x*sin(x)*sin(n*x) n *cos(x)*sin(n*x) 2*n*cos(n*x)*sin(x) n*x*cos(x)*cos(n*x) x*n *sin(x)*sin(n*x) x*n *cos(x)*cos(n*x) |
||--------------- + ----------------- + ------------------ - ------------------- + ------------------- - -------------------- - -------------------- otherwise |
|| 4 2 4 2 4 2 4 2 4 2 4 2 4 2 |
\\ 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n /
∫xcos(x)sin(nx)dx=C+⎩⎨⎧−4xsin2(x)+4xcos2(x)−4sin(x)cos(x)4xsin2(x)−4xcos2(x)+4sin(x)cos(x)−n4−2n2+1n3xcos(x)cos(nx)−n4−2n2+1n2xsin(x)sin(nx)+n4−2n2+1n2sin(nx)cos(x)+n4−2n2+1nxcos(x)cos(nx)−n4−2n2+12nsin(x)cos(nx)+n4−2n2+1xsin(x)sin(nx)+n4−2n2+1sin(nx)cos(x)forn=−1forn=1otherwise
/ pi
| -- for n = -1
| 4
|
| -pi
| ---- for n = 1
< 4
|
| 2 3
| sin(pi*n) n *sin(pi*n) pi*n *cos(pi*n) pi*n*cos(pi*n)
|- ------------- - ------------- + --------------- - -------------- otherwise
| 4 2 4 2 4 2 4 2
\ 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n
⎩⎨⎧4π−4πn4−2n2+1πn3cos(πn)−n4−2n2+1n2sin(πn)−n4−2n2+1πncos(πn)−n4−2n2+1sin(πn)forn=−1forn=1otherwise
=
/ pi
| -- for n = -1
| 4
|
| -pi
| ---- for n = 1
< 4
|
| 2 3
| sin(pi*n) n *sin(pi*n) pi*n *cos(pi*n) pi*n*cos(pi*n)
|- ------------- - ------------- + --------------- - -------------- otherwise
| 4 2 4 2 4 2 4 2
\ 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n 1 + n - 2*n
⎩⎨⎧4π−4πn4−2n2+1πn3cos(πn)−n4−2n2+1n2sin(πn)−n4−2n2+1πncos(πn)−n4−2n2+1sin(πn)forn=−1forn=1otherwise
Piecewise((pi/4, n = -1), (-pi/4, n = 1), (-sin(pi*n)/(1 + n^4 - 2*n^2) - n^2*sin(pi*n)/(1 + n^4 - 2*n^2) + pi*n^3*cos(pi*n)/(1 + n^4 - 2*n^2) - pi*n*cos(pi*n)/(1 + n^4 - 2*n^2), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.