Sr Examen
Integral de xsin(x/2)cos(kx) dx
Solución
Ha introducido
[src]
2*pi / | | /x\ | x*sin|-|*cos(k*x) dx | \2/ | / 0
$$\int\limits_{0}^{2 \pi} x \sin{\left(\frac{x}{2} \right)} \cos{\left(k x \right)}\, dx$$
Integral((x*sin(x/2))*cos(k*x), (x, 0, 2*pi))
Respuesta
[src]
/ -pi for Or(k = -1/2, k = 1/2) | | 2 <4*pi*cos(2*pi*k) 16*k*sin(2*pi*k) 16*pi*k *cos(2*pi*k) |---------------- + ---------------- - -------------------- otherwise | 2 4 2 4 2 4 \1 - 8*k + 16*k 1 - 8*k + 16*k 1 - 8*k + 16*k
$$\begin{cases} - \pi & \text{for}\: k = - \frac{1}{2} \vee k = \frac{1}{2} \\- \frac{16 \pi k^{2} \cos{\left(2 \pi k \right)}}{16 k^{4} - 8 k^{2} + 1} + \frac{16 k \sin{\left(2 \pi k \right)}}{16 k^{4} - 8 k^{2} + 1} + \frac{4 \pi \cos{\left(2 \pi k \right)}}{16 k^{4} - 8 k^{2} + 1} & \text{otherwise} \end{cases}$$
=
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/ -pi for Or(k = -1/2, k = 1/2) | | 2 <4*pi*cos(2*pi*k) 16*k*sin(2*pi*k) 16*pi*k *cos(2*pi*k) |---------------- + ---------------- - -------------------- otherwise | 2 4 2 4 2 4 \1 - 8*k + 16*k 1 - 8*k + 16*k 1 - 8*k + 16*k
$$\begin{cases} - \pi & \text{for}\: k = - \frac{1}{2} \vee k = \frac{1}{2} \\- \frac{16 \pi k^{2} \cos{\left(2 \pi k \right)}}{16 k^{4} - 8 k^{2} + 1} + \frac{16 k \sin{\left(2 \pi k \right)}}{16 k^{4} - 8 k^{2} + 1} + \frac{4 \pi \cos{\left(2 \pi k \right)}}{16 k^{4} - 8 k^{2} + 1} & \text{otherwise} \end{cases}$$
Piecewise((-pi, (k = -1/2)∨(k = 1/2)), (4*pi*cos(2*pi*k)/(1 - 8*k^2 + 16*k^4) + 16*k*sin(2*pi*k)/(1 - 8*k^2 + 16*k^4) - 16*pi*k^2*cos(2*pi*k)/(1 - 8*k^2 + 16*k^4), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.