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Solving integrals/ x*sin(m*x/a)

Integral of x*sin(m*x/a) dx

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The solution

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

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

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