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Integral of 12*y*sin(2*x*y) dy

Limits of integration:

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The solution

You have entered [src]
 pi                   
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 |  12*y*sin(2*x*y) dy
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pi                    
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$$\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{2}} 12 y \sin{\left(2 x y \right)}\, dy$$
Integral((12*y)*sin((2*x)*y), (y, pi/4, pi/2))
The answer (Indefinite) [src]
                               //              0                for x = 0\                                  
                               ||                                        |                                  
  /                            || //sin(2*x*y)              \            |        //     0        for x = 0\
 |                             || ||----------  for 2*x != 0|            |        ||                       |
 | 12*y*sin(2*x*y) dy = C - 12*|<-|<   2*x                  |            | + 12*y*|<-cos(2*x*y)            |
 |                             || ||                        |            |        ||------------  otherwise|
/                              || \\    y        otherwise  /            |        \\    2*x                /
                               ||-----------------------------  otherwise|                                  
                               \\             2*x                        /                                  
$$\int 12 y \sin{\left(2 x y \right)}\, dy = C + 12 y \left(\begin{cases} 0 & \text{for}\: x = 0 \\- \frac{\cos{\left(2 x y \right)}}{2 x} & \text{otherwise} \end{cases}\right) - 12 \left(\begin{cases} 0 & \text{for}\: x = 0 \\- \frac{\begin{cases} \frac{\sin{\left(2 x y \right)}}{2 x} & \text{for}\: 2 x \neq 0 \\y & \text{otherwise} \end{cases}}{2 x} & \text{otherwise} \end{cases}\right)$$
The answer [src]
/       /pi*x\                                          /pi*x\                                  
|  3*sin|----|                                  3*pi*cos|----|                                  
|       \ 2  /   3*sin(pi*x)   3*pi*cos(pi*x)           \ 2  /                                  
|- ----------- + ----------- - -------------- + --------------  for And(x > -oo, x < oo, x != 0)
<        2             2             x               2*x                                        
|       x             x                                                                         
|                                                                                               
|                              0                                           otherwise            
\                                                                                               
$$\begin{cases} \frac{3 \pi \cos{\left(\frac{\pi x}{2} \right)}}{2 x} - \frac{3 \pi \cos{\left(\pi x \right)}}{x} - \frac{3 \sin{\left(\frac{\pi x}{2} \right)}}{x^{2}} + \frac{3 \sin{\left(\pi x \right)}}{x^{2}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
=
/       /pi*x\                                          /pi*x\                                  
|  3*sin|----|                                  3*pi*cos|----|                                  
|       \ 2  /   3*sin(pi*x)   3*pi*cos(pi*x)           \ 2  /                                  
|- ----------- + ----------- - -------------- + --------------  for And(x > -oo, x < oo, x != 0)
<        2             2             x               2*x                                        
|       x             x                                                                         
|                                                                                               
|                              0                                           otherwise            
\                                                                                               
$$\begin{cases} \frac{3 \pi \cos{\left(\frac{\pi x}{2} \right)}}{2 x} - \frac{3 \pi \cos{\left(\pi x \right)}}{x} - \frac{3 \sin{\left(\frac{\pi x}{2} \right)}}{x^{2}} + \frac{3 \sin{\left(\pi x \right)}}{x^{2}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((-3*sin(pi*x/2)/x^2 + 3*sin(pi*x)/x^2 - 3*pi*cos(pi*x)/x + 3*pi*cos(pi*x/2)/(2*x), (x > -oo)∧(x < oo)∧(Ne(x, 0))), (0, True))

    Use the examples entering the upper and lower limits of integration.