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