Sr Examen
Integral de (1/2)^(1/2)*(6sin(4pix))sin(pi*n*x/4) dx
Solución
Ha introducido
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4 / | | 1 /pi*n*x\ | -----*6*sin(4*pi*x)*sin|------| dx | ___ \ 4 / | \/ 2 | / 0
$$\int\limits_{0}^{4} \frac{6 \sin{\left(4 \pi x \right)}}{\sqrt{2}} \sin{\left(\frac{x \pi n}{4} \right)}\, dx$$
Integral((sqrt(1/2)*(6*sin((4*pi)*x)))*sin(((pi*n)*x)/4), (x, 0, 4))
Respuesta (Indefinida)
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// / pi*n*x\ / / pi*n*x\ \
|| 8*tan|-2*pi*x + ------| | 8*tan|2*pi*x + ------| |
|| \ 8 / | \ 8 / |
||-------------------------------------------------------------------------- for n != 16 |----------------------------------------------------------------------- for n != -16|
|< 2/ pi*n*x\ 2/ pi*n*x\ < 2/ pi*n*x\ 2/ pi*n*x\ |
||-16*pi + pi*n - 16*pi*tan |-2*pi*x + ------| + pi*n*tan |-2*pi*x + ------| |16*pi + pi*n + 16*pi*tan |2*pi*x + ------| + pi*n*tan |2*pi*x + ------| |
/ || \ 8 / \ 8 / | \ 8 / \ 8 / |
| || | |
| 1 /pi*n*x\ ___ |\ x otherwise \ x otherwise |
| -----*6*sin(4*pi*x)*sin|------| dx = C + 3*\/ 2 *|---------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------|
| ___ \ 4 / \ 2 2 /
| \/ 2
|
/
$$\int \frac{6 \sin{\left(4 \pi x \right)}}{\sqrt{2}} \sin{\left(\frac{x \pi n}{4} \right)}\, dx = C + 3 \sqrt{2} \left(\frac{\begin{cases} \frac{8 \tan{\left(\frac{\pi n x}{8} - 2 \pi x \right)}}{\pi n \tan^{2}{\left(\frac{\pi n x}{8} - 2 \pi x \right)} + \pi n - 16 \pi \tan^{2}{\left(\frac{\pi n x}{8} - 2 \pi x \right)} - 16 \pi} & \text{for}\: n \neq 16 \\x & \text{otherwise} \end{cases}}{2} - \frac{\begin{cases} \frac{8 \tan{\left(\frac{\pi n x}{8} + 2 \pi x \right)}}{\pi n \tan^{2}{\left(\frac{\pi n x}{8} + 2 \pi x \right)} + \pi n + 16 \pi \tan^{2}{\left(\frac{\pi n x}{8} + 2 \pi x \right)} + 16 \pi} & \text{for}\: n \neq -16 \\x & \text{otherwise} \end{cases}}{2}\right)$$
Respuesta
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/ ___ | -6*\/ 2 for n = -16 | | ___ | 6*\/ 2 for n = 16 < | ___ |192*\/ 2 *sin(pi*n) |------------------- otherwise | 2 \ -256*pi + pi*n
$$\begin{cases} - 6 \sqrt{2} & \text{for}\: n = -16 \\6 \sqrt{2} & \text{for}\: n = 16 \\\frac{192 \sqrt{2} \sin{\left(\pi n \right)}}{\pi n^{2} - 256 \pi} & \text{otherwise} \end{cases}$$
=
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/ ___ | -6*\/ 2 for n = -16 | | ___ | 6*\/ 2 for n = 16 < | ___ |192*\/ 2 *sin(pi*n) |------------------- otherwise | 2 \ -256*pi + pi*n
$$\begin{cases} - 6 \sqrt{2} & \text{for}\: n = -16 \\6 \sqrt{2} & \text{for}\: n = 16 \\\frac{192 \sqrt{2} \sin{\left(\pi n \right)}}{\pi n^{2} - 256 \pi} & \text{otherwise} \end{cases}$$
Piecewise((-6*sqrt(2), n = -16), (6*sqrt(2), n = 16), (192*sqrt(2)*sin(pi*n)/(-256*pi + pi*n^2), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.