Sr Examen
Integral de e^(-x/4)sinnx dx
Solución
Ha introducido
[src]
pi / | | -x | --- | 4 | E *sin(n*x) dx | / 0
$$\int\limits_{0}^{\pi} e^{\frac{\left(-1\right) x}{4}} \sin{\left(n x \right)}\, dx$$
Integral(E^((-x)/4)*sin(n*x), (x, 0, pi))
Respuesta (Indefinida)
[src]
// / -x -x \ \
|| | --- --- | |
|| | -x /x\ 4 4 /x\| |
|| | --- x*cosh|-|*e x*e *sinh|-|| |
|| | /x\ 4 \4/ \4/| -I |
||-I*|2*cosh|-|*e + -------------- + --------------| for n = ---|
/ || \ \4/ 2 2 / 4 |
| || |
| -x || / -x -x \ |
| --- || | --- --- | |
| 4 || | -x /x\ 4 4 /x\| |
| E *sin(n*x) dx = C + |< | --- x*cosh|-|*e x*e *sinh|-|| |
| || | /x\ 4 \4/ \4/| I |
/ ||I*|2*cosh|-|*e + -------------- + --------------| for n = - |
|| \ \4/ 2 2 / 4 |
|| |
|| 4*sin(n*x) 16*n*cos(n*x) |
|| - ------------- - ------------- otherwise |
|| x x x x |
|| - - - - |
|| 2 4 4 2 4 4 |
|| 16*n *e + e 16*n *e + e |
\\ /
$$\int e^{\frac{\left(-1\right) x}{4}} \sin{\left(n x \right)}\, dx = C + \begin{cases} - i \left(\frac{x e^{- \frac{x}{4}} \sinh{\left(\frac{x}{4} \right)}}{2} + \frac{x e^{- \frac{x}{4}} \cosh{\left(\frac{x}{4} \right)}}{2} + 2 e^{- \frac{x}{4}} \cosh{\left(\frac{x}{4} \right)}\right) & \text{for}\: n = - \frac{i}{4} \\i \left(\frac{x e^{- \frac{x}{4}} \sinh{\left(\frac{x}{4} \right)}}{2} + \frac{x e^{- \frac{x}{4}} \cosh{\left(\frac{x}{4} \right)}}{2} + 2 e^{- \frac{x}{4}} \cosh{\left(\frac{x}{4} \right)}\right) & \text{for}\: n = \frac{i}{4} \\- \frac{16 n \cos{\left(n x \right)}}{16 n^{2} e^{\frac{x}{4}} + e^{\frac{x}{4}}} - \frac{4 \sin{\left(n x \right)}}{16 n^{2} e^{\frac{x}{4}} + e^{\frac{x}{4}}} & \text{otherwise} \end{cases}$$
Respuesta
[src]
4*sin(pi*n) 16*n 16*n*cos(pi*n)
- --------------- + --------- - ---------------
pi pi 2 pi pi
-- -- 1 + 16*n -- --
2 4 4 2 4 4
16*n *e + e 16*n *e + e
$$- \frac{16 n \cos{\left(\pi n \right)}}{16 n^{2} e^{\frac{\pi}{4}} + e^{\frac{\pi}{4}}} + \frac{16 n}{16 n^{2} + 1} - \frac{4 \sin{\left(\pi n \right)}}{16 n^{2} e^{\frac{\pi}{4}} + e^{\frac{\pi}{4}}}$$
=
=
4*sin(pi*n) 16*n 16*n*cos(pi*n)
- --------------- + --------- - ---------------
pi pi 2 pi pi
-- -- 1 + 16*n -- --
2 4 4 2 4 4
16*n *e + e 16*n *e + e
$$- \frac{16 n \cos{\left(\pi n \right)}}{16 n^{2} e^{\frac{\pi}{4}} + e^{\frac{\pi}{4}}} + \frac{16 n}{16 n^{2} + 1} - \frac{4 \sin{\left(\pi n \right)}}{16 n^{2} e^{\frac{\pi}{4}} + e^{\frac{\pi}{4}}}$$
-4*sin(pi*n)/(16*n^2*exp(pi/4) + exp(pi/4)) + 16*n/(1 + 16*n^2) - 16*n*cos(pi*n)/(16*n^2*exp(pi/4) + exp(pi/4))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.