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