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Resolución de integrales/ y*cos(2*x*y)

Integral de y*cos(2*x*y) dy

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 pi                
  /                
 |                 
 |  y*cos(2*x*y) dy
 |                 
/                  
pi                 
--                 
2
$$\int\limits_{\frac{\pi}{2}}^{\pi} y \cos{\left(2 x y \right)}\, dy$$ 
Integral(y*cos((2*x)*y), (y, pi/2, pi))
Respuesta (Indefinida) [src] 
                         //             2                        \                             
                         ||            y                         |                             
                         ||            --               for x = 0|                             
                         ||            2                         |                             
  /                      ||                                      |     //    y       for x = 0\
 |                       ||/-cos(2*x*y)                          |     ||                     |
 | y*cos(2*x*y) dy = C - |<|------------  for 2*x != 0           | + y*|
$$\int y \cos{\left(2 x y \right)}\, dy = C + y \left(\begin{cases} y & \text{for}\: x = 0 \\\frac{\sin{\left(2 x y \right)}}{2 x} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{y^{2}}{2} & \text{for}\: x = 0 \\\frac{\begin{cases} - \frac{\cos{\left(2 x y \right)}}{2 x} & \text{for}\: 2 x \neq 0 \\0 & \text{otherwise} \end{cases}}{2 x} & \text{otherwise} \end{cases}$$
Simplificar
Respuesta [src] 
/  cos(pi*x)   cos(2*pi*x)   pi*sin(2*pi*x)   pi*sin(pi*x)                                  
|- --------- + ----------- + -------------- - ------------  for And(x > -oo, x < oo, x != 0)
|        2            2           2*x             4*x                                       
|     4*x          4*x                                                                      
<                                                                                           
|                              2                                                            
|                          3*pi                                                             
|                          -----                                       otherwise            
\                            8
$$\begin{cases} - \frac{\pi \sin{\left(\pi x \right)}}{4 x} + \frac{\pi \sin{\left(2 \pi x \right)}}{2 x} - \frac{\cos{\left(\pi x \right)}}{4 x^{2}} + \frac{\cos{\left(2 \pi x \right)}}{4 x^{2}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\\frac{3 \pi^{2}}{8} & \text{otherwise} \end{cases}$$ 
=
= 
/  cos(pi*x)   cos(2*pi*x)   pi*sin(2*pi*x)   pi*sin(pi*x)                                  
|- --------- + ----------- + -------------- - ------------  for And(x > -oo, x < oo, x != 0)
|        2            2           2*x             4*x                                       
|     4*x          4*x                                                                      
<                                                                                           
|                              2                                                            
|                          3*pi                                                             
|                          -----                                       otherwise            
\                            8
$$\begin{cases} - \frac{\pi \sin{\left(\pi x \right)}}{4 x} + \frac{\pi \sin{\left(2 \pi x \right)}}{2 x} - \frac{\cos{\left(\pi x \right)}}{4 x^{2}} + \frac{\cos{\left(2 \pi x \right)}}{4 x^{2}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\\frac{3 \pi^{2}}{8} & \text{otherwise} \end{cases}$$ 
Piecewise((-cos(pi*x)/(4*x^2) + cos(2*pi*x)/(4*x^2) + pi*sin(2*pi*x)/(2*x) - pi*sin(pi*x)/(4*x), (x > -oo)∧(x < oo)∧(Ne(x, 0))), (3*pi^2/8, True))

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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