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Solving integrals/ ycos(xy)

Integral of ycos(xy) dy

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The solution

You have entered [src] 
 pi              
  /              
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 |  y*cos(x*y) dy
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0
∫0πycos⁡(xy) dy\int\limits_{0}^{\pi} y \cos{\left(x y \right)}\, dy 
Integral(y*cos(x*y), (y, 0, pi))
The answer (Indefinite) [src] 
                       //           2                      \                           
                       ||          y                       |                           
                       ||          --             for x = 0|                           
                       ||          2                       |                           
  /                    ||                                  |     //   y      for x = 0\
 |                     ||/-cos(x*y)                        |     ||                   |
 | y*cos(x*y) dy = C - |<|----------  for x != 0           | + y*|
∫ycos⁡(xy) dy=C+y({yfor x=0sin⁡(xy)xotherwise)−{y22for x=0{−cos⁡(xy)xfor x≠00otherwisexotherwise\int y \cos{\left(x y \right)}\, dy = C + y \left(\begin{cases} y & \text{for}\: x = 0 \\\frac{\sin{\left(x y \right)}}{x} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{y^{2}}{2} & \text{for}\: x = 0 \\\frac{\begin{cases} - \frac{\cos{\left(x y \right)}}{x} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}}{x} & \text{otherwise} \end{cases}
Simplify
The answer [src] 
/  1    cos(pi*x)   pi*sin(pi*x)                                  
|- -- + --------- + ------------  for And(x > -oo, x < oo, x != 0)
|   2        2           x                                        
|  x        x                                                     
<                                                                 
|                2                                                
|              pi                                                 
|              ---                           otherwise            
\               2
{πsin⁡(πx)x+cos⁡(πx)x2−1x2for x>−∞∧x<∞∧x≠0π22otherwise\begin{cases} \frac{\pi \sin{\left(\pi x \right)}}{x} + \frac{\cos{\left(\pi x \right)}}{x^{2}} - \frac{1}{x^{2}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\\frac{\pi^{2}}{2} & \text{otherwise} \end{cases} 
=
= 
/  1    cos(pi*x)   pi*sin(pi*x)                                  
|- -- + --------- + ------------  for And(x > -oo, x < oo, x != 0)
|   2        2           x                                        
|  x        x                                                     
<                                                                 
|                2                                                
|              pi                                                 
|              ---                           otherwise            
\               2
{πsin⁡(πx)x+cos⁡(πx)x2−1x2for x>−∞∧x<∞∧x≠0π22otherwise\begin{cases} \frac{\pi \sin{\left(\pi x \right)}}{x} + \frac{\cos{\left(\pi x \right)}}{x^{2}} - \frac{1}{x^{2}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\\frac{\pi^{2}}{2} & \text{otherwise} \end{cases} 
Piecewise((-1/x^2 + cos(pi*x)/x^2 + pi*sin(pi*x)/x, (x > -oo)∧(x < oo)∧(Ne(x, 0))), (pi^2/2, True))

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

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