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Solving integrals/ y*sin(2xy)

Integral of y*sin(2xy) dy

Limits of integration:

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Piecewise:

The solution

You have entered [src] 
  1                
  /                
 |                 
 |  y*sin(2*x*y) dy
 |                 
/                  
0
$$\int\limits_{0}^{1} y \sin{\left(2 x y \right)}\, dy$$ 
Integral(y*sin((2*x)*y), (y, 0, 1))
The answer (Indefinite) [src] 
                         //              0                for x = 0\                               
                         ||                                        |                               
  /                      || //sin(2*x*y)              \            |     //     0        for x = 0\
 |                       || ||----------  for 2*x != 0|            |     ||                       |
 | y*sin(2*x*y) dy = C - |<-|<   2*x                  |            | + y*|<-cos(2*x*y)            |
 |                       || ||                        |            |     ||------------  otherwise|
/                        || \\    y        otherwise  /            |     \\    2*x                /
                         ||-----------------------------  otherwise|                               
                         \\             2*x                        /
$$\int y \sin{\left(2 x y \right)}\, dy = C + y \left(\begin{cases} 0 & \text{for}\: x = 0 \\- \frac{\cos{\left(2 x y \right)}}{2 x} & \text{otherwise} \end{cases}\right) - \begin{cases} 0 & \text{for}\: x = 0 \\- \frac{\begin{cases} \frac{\sin{\left(2 x y \right)}}{2 x} & \text{for}\: 2 x \neq 0 \\y & \text{otherwise} \end{cases}}{2 x} & \text{otherwise} \end{cases}$$
Simplify
The answer [src] 
/  cos(2*x)   sin(2*x)                                  
|- -------- + --------  for And(x > -oo, x < oo, x != 0)
|    2*x           2                                    
<               4*x                                     
|                                                       
|          0                       otherwise            
\
$$\begin{cases} - \frac{\cos{\left(2 x \right)}}{2 x} + \frac{\sin{\left(2 x \right)}}{4 x^{2}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
=
= 
/  cos(2*x)   sin(2*x)                                  
|- -------- + --------  for And(x > -oo, x < oo, x != 0)
|    2*x           2                                    
<               4*x                                     
|                                                       
|          0                       otherwise            
\
$$\begin{cases} - \frac{\cos{\left(2 x \right)}}{2 x} + \frac{\sin{\left(2 x \right)}}{4 x^{2}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
Piecewise((-cos(2*x)/(2*x) + sin(2*x)/(4*x^2), (x > -oo)∧(x < oo)∧(Ne(x, 0))), (0, True))

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

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