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