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