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