Sr Examen
Integral de (2)3sinxsin(npix)dx dx
Solución
Ha introducido
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1 / | | 6*sin(x)*sin(n*pi*x) dx | / 0
$$\int\limits_{0}^{1} 6 \sin{\left(x \right)} \sin{\left(x \pi n \right)}\, dx$$
Integral((6*sin(x))*sin((n*pi)*x), (x, 0, 1))
Respuesta (Indefinida)
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// /x pi*n*x\ \ // / x pi*n*x\ \
|| 2*tan|- + ------| | || 2*tan|- - + ------| |
/ || \2 2 / -1 | || \ 2 2 / 1 |
| ||--------------------------------------------------- for n != ---| ||-------------------------------------------------------- for n != --|
| 6*sin(x)*sin(n*pi*x) dx = C - 3*|< 2/x pi*n*x\ 2/x pi*n*x\ pi| + 3*|< 2/ x pi*n*x\ 2/ x pi*n*x\ pi|
| ||1 + tan |- + ------| + pi*n + pi*n*tan |- + ------| | ||-1 - tan |- - + ------| + pi*n + pi*n*tan |- - + ------| |
/ || \2 2 / \2 2 / | || \ 2 2 / \ 2 2 / |
|| | || |
\\ x otherwise / \\ x otherwise /
$$\int 6 \sin{\left(x \right)} \sin{\left(x \pi n \right)}\, dx = C + 3 \left(\begin{cases} \frac{2 \tan{\left(\frac{\pi n x}{2} - \frac{x}{2} \right)}}{\pi n \tan^{2}{\left(\frac{\pi n x}{2} - \frac{x}{2} \right)} + \pi n - \tan^{2}{\left(\frac{\pi n x}{2} - \frac{x}{2} \right)} - 1} & \text{for}\: n \neq \frac{1}{\pi} \\x & \text{otherwise} \end{cases}\right) - 3 \left(\begin{cases} \frac{2 \tan{\left(\frac{\pi n x}{2} + \frac{x}{2} \right)}}{\pi n \tan^{2}{\left(\frac{\pi n x}{2} + \frac{x}{2} \right)} + \pi n + \tan^{2}{\left(\frac{\pi n x}{2} + \frac{x}{2} \right)} + 1} & \text{for}\: n \neq - \frac{1}{\pi} \\x & \text{otherwise} \end{cases}\right)$$
Respuesta
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/ 2 2 1 | 3*cos (1) + 3*sin (1) - 3*cos(1)*sin(1) for n = -- | pi | | 2 2 -1 | - 3*cos (1) - 3*sin (1) + 3*cos(1)*sin(1) for n = --- < pi | |6*cos(1)*sin(pi*n) 6*pi*n*cos(pi*n)*sin(1) |------------------ - ----------------------- otherwise | 2 2 2 2 | -1 + pi *n -1 + pi *n \
$$\begin{cases} - 3 \sin{\left(1 \right)} \cos{\left(1 \right)} + 3 \cos^{2}{\left(1 \right)} + 3 \sin^{2}{\left(1 \right)} & \text{for}\: n = \frac{1}{\pi} \\- 3 \sin^{2}{\left(1 \right)} - 3 \cos^{2}{\left(1 \right)} + 3 \sin{\left(1 \right)} \cos{\left(1 \right)} & \text{for}\: n = - \frac{1}{\pi} \\- \frac{6 \pi n \sin{\left(1 \right)} \cos{\left(\pi n \right)}}{\pi^{2} n^{2} - 1} + \frac{6 \sin{\left(\pi n \right)} \cos{\left(1 \right)}}{\pi^{2} n^{2} - 1} & \text{otherwise} \end{cases}$$
=
=
/ 2 2 1 | 3*cos (1) + 3*sin (1) - 3*cos(1)*sin(1) for n = -- | pi | | 2 2 -1 | - 3*cos (1) - 3*sin (1) + 3*cos(1)*sin(1) for n = --- < pi | |6*cos(1)*sin(pi*n) 6*pi*n*cos(pi*n)*sin(1) |------------------ - ----------------------- otherwise | 2 2 2 2 | -1 + pi *n -1 + pi *n \
$$\begin{cases} - 3 \sin{\left(1 \right)} \cos{\left(1 \right)} + 3 \cos^{2}{\left(1 \right)} + 3 \sin^{2}{\left(1 \right)} & \text{for}\: n = \frac{1}{\pi} \\- 3 \sin^{2}{\left(1 \right)} - 3 \cos^{2}{\left(1 \right)} + 3 \sin{\left(1 \right)} \cos{\left(1 \right)} & \text{for}\: n = - \frac{1}{\pi} \\- \frac{6 \pi n \sin{\left(1 \right)} \cos{\left(\pi n \right)}}{\pi^{2} n^{2} - 1} + \frac{6 \sin{\left(\pi n \right)} \cos{\left(1 \right)}}{\pi^{2} n^{2} - 1} & \text{otherwise} \end{cases}$$
Piecewise((3*cos(1)^2 + 3*sin(1)^2 - 3*cos(1)*sin(1), n = 1/pi), (-3*cos(1)^2 - 3*sin(1)^2 + 3*cos(1)*sin(1), n = -1/pi), (6*cos(1)*sin(pi*n)/(-1 + pi^2*n^2) - 6*pi*n*cos(pi*n)*sin(1)/(-1 + pi^2*n^2), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.