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Integral of sin(ax)*cos(bx) dx

Limits of integration:

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Piecewise:

The solution

You have entered [src]
  1                     
  /                     
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 |  sin(a*x)*cos(b*x) dx
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0                       
$$\int\limits_{0}^{1} \sin{\left(a x \right)} \cos{\left(b x \right)}\, dx$$
The answer (Indefinite) [src]
                              //                     0                       for And(a = 0, b = 0)\
                              ||                                                                  |
                              ||                    2                                             |
                              ||                 cos (b*x)                                        |
                              ||                 ---------                        for a = -b      |
                              ||                    2*b                                           |
  /                           ||                                                                  |
 |                            ||                    2                                             |
 | sin(a*x)*cos(b*x) dx = C + |<                -cos (b*x)                                        |
 |                            ||                -----------                        for a = b      |
/                             ||                    2*b                                           |
                              ||                                                                  |
                              ||  a*cos(a*x)*cos(b*x)   b*sin(a*x)*sin(b*x)                       |
                              ||- ------------------- - -------------------        otherwise      |
                              ||         2    2                2    2                             |
                              ||        a  - b                a  - b                              |
                              \\                                                                  /
$$-{{\cos \left(\left(b+a\right)\,x\right)}\over{2\,\left(b+a\right) }}-{{\cos \left(\left(a-b\right)\,x\right)}\over{2\,\left(a-b\right) }}$$
The answer [src]
/                     0                               for Or(And(a = 0, b = 0), And(a = 0, a = b, b = 0), And(a = 0, a = -b, b = 0), And(a = 0, a = -b, a = b, b = 0))        
|                                                                                                                                                                             
|                         2                                                                                                                                                   
|                 1    cos (b)                                                                                                                                                
|              - --- + -------                for Or(And(a = 0, a = -b), And(a = -b, a = b), And(a = -b, b = 0), And(a = 0, a = -b, a = b), And(a = -b, a = b, b = 0), a = -b)
|                2*b     2*b                                                                                                                                                  
|                                                                                                                                                                             
|                        2                                                                                                                                                    
<                1    cos (b)                                                                                                                                                 
|               --- - -------                                                       for Or(And(a = 0, a = b), And(a = b, b = 0), a = b)                                       
|               2*b     2*b                                                                                                                                                   
|                                                                                                                                                                             
|   a      a*cos(a)*cos(b)   b*sin(a)*sin(b)                                                                                                                                  
|------- - --------------- - ---------------                                                             otherwise                                                            
| 2    2        2    2            2    2                                                                                                                                      
|a  - b        a  - b            a  - b                                                                                                                                       
\                                                                                                                                                                             
$$-{{\left(b-a\right)\,\cos \left(b+a\right)+\left(-b-a\right)\,\cos \left(b-a\right)}\over{2\,b^2-2\,a^2}}-{{a}\over{b^2-a^2}}$$
=
=
/                     0                               for Or(And(a = 0, b = 0), And(a = 0, a = b, b = 0), And(a = 0, a = -b, b = 0), And(a = 0, a = -b, a = b, b = 0))        
|                                                                                                                                                                             
|                         2                                                                                                                                                   
|                 1    cos (b)                                                                                                                                                
|              - --- + -------                for Or(And(a = 0, a = -b), And(a = -b, a = b), And(a = -b, b = 0), And(a = 0, a = -b, a = b), And(a = -b, a = b, b = 0), a = -b)
|                2*b     2*b                                                                                                                                                  
|                                                                                                                                                                             
|                        2                                                                                                                                                    
<                1    cos (b)                                                                                                                                                 
|               --- - -------                                                       for Or(And(a = 0, a = b), And(a = b, b = 0), a = b)                                       
|               2*b     2*b                                                                                                                                                   
|                                                                                                                                                                             
|   a      a*cos(a)*cos(b)   b*sin(a)*sin(b)                                                                                                                                  
|------- - --------------- - ---------------                                                             otherwise                                                            
| 2    2        2    2            2    2                                                                                                                                      
|a  - b        a  - b            a  - b                                                                                                                                       
\                                                                                                                                                                             
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b \wedge b = 0\right) \\\frac{\cos^{2}{\left(b \right)}}{2 b} - \frac{1}{2 b} & \text{for}\: \left(a = 0 \wedge a = - b\right) \vee \left(a = - b \wedge a = b\right) \vee \left(a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b\right) \vee \left(a = - b \wedge a = b \wedge b = 0\right) \vee a = - b \\- \frac{\cos^{2}{\left(b \right)}}{2 b} + \frac{1}{2 b} & \text{for}\: \left(a = 0 \wedge a = b\right) \vee \left(a = b \wedge b = 0\right) \vee a = b \\- \frac{a \cos{\left(a \right)} \cos{\left(b \right)}}{a^{2} - b^{2}} - \frac{b \sin{\left(a \right)} \sin{\left(b \right)}}{a^{2} - b^{2}} + \frac{a}{a^{2} - b^{2}} & \text{otherwise} \end{cases}$$

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