Sr Examen

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  1                     
  /                     
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 |  sin(a*x)*cos(b*x) dx
 |                      
/                       
0                       
$$\int\limits_{0}^{1} \sin{\left(a x \right)} \cos{\left(b x \right)}\, dx$$
Integral(sin(a*x)*cos(b*x), (x, 0, 1))
Respuesta (Indefinida) [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                              |
                              \\                                                                  /
$$\int \sin{\left(a x \right)} \cos{\left(b x \right)}\, dx = C + \begin{cases} 0 & \text{for}\: a = 0 \wedge b = 0 \\\frac{\cos^{2}{\left(b x \right)}}{2 b} & \text{for}\: a = - b \\- \frac{\cos^{2}{\left(b x \right)}}{2 b} & \text{for}\: a = b \\- \frac{a \cos{\left(a x \right)} \cos{\left(b x \right)}}{a^{2} - b^{2}} - \frac{b \sin{\left(a x \right)} \sin{\left(b x \right)}}{a^{2} - b^{2}} & \text{otherwise} \end{cases}$$
Respuesta [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                                                                                                                                                       
|                 -sin (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                                                                                                                                                       
<                  sin (b)                                                                                                                                                    
|                  -------                                                          for Or(And(a = 0, a = b), And(a = b, b = 0), a = 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                                                                                                                                       
\                                                                                                                                                                             
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b \wedge b = 0\right) \\- \frac{\sin^{2}{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = - b\right) \vee \left(a = - b \wedge a = b\right) \vee \left(a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b\right) \vee \left(a = - b \wedge a = b \wedge b = 0\right) \vee a = - b \\\frac{\sin^{2}{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = b\right) \vee \left(a = b \wedge b = 0\right) \vee a = b \\- \frac{a \cos{\left(a \right)} \cos{\left(b \right)}}{a^{2} - b^{2}} + \frac{a}{a^{2} - b^{2}} - \frac{b \sin{\left(a \right)} \sin{\left(b \right)}}{a^{2} - b^{2}} & \text{otherwise} \end{cases}$$
=
=
/                     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                                                                                                                                                       
|                 -sin (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                                                                                                                                                       
<                  sin (b)                                                                                                                                                    
|                  -------                                                          for Or(And(a = 0, a = b), And(a = b, b = 0), a = 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                                                                                                                                       
\                                                                                                                                                                             
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b \wedge b = 0\right) \\- \frac{\sin^{2}{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = - b\right) \vee \left(a = - b \wedge a = b\right) \vee \left(a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b\right) \vee \left(a = - b \wedge a = b \wedge b = 0\right) \vee a = - b \\\frac{\sin^{2}{\left(b \right)}}{2 b} & \text{for}\: \left(a = 0 \wedge a = b\right) \vee \left(a = b \wedge b = 0\right) \vee a = b \\- \frac{a \cos{\left(a \right)} \cos{\left(b \right)}}{a^{2} - b^{2}} + \frac{a}{a^{2} - b^{2}} - \frac{b \sin{\left(a \right)} \sin{\left(b \right)}}{a^{2} - b^{2}} & \text{otherwise} \end{cases}$$
Piecewise((0, ((a = 0)∧(b = 0))∨((a = 0)∧(a = b)∧(b = 0))∨((a = 0)∧(b = 0)∧(a = -b))∨((a = 0)∧(a = b)∧(b = 0)∧(a = -b))), (-sin(b)^2/(2*b), (a = -b)∨((a = 0)∧(a = -b))∨((a = b)∧(a = -b))∨((b = 0)∧(a = -b))∨((a = 0)∧(a = b)∧(a = -b))∨((a = b)∧(b = 0)∧(a = -b))), (sin(b)^2/(2*b), (a = b)∨((a = 0)∧(a = b))∨((a = b)∧(b = 0))), (a/(a^2 - b^2) - a*cos(a)*cos(b)/(a^2 - b^2) - b*sin(a)*sin(b)/(a^2 - b^2), True))

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.