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