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Resolución de integrales/ cos(b*x)sin(a*x)

Integral de cos(b*x)sin(a*x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  c                     
  /                     
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 |  cos(b*x)*sin(a*x) dx
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/                       
0
∫0csin⁡(ax)cos⁡(bx) dx\int\limits_{0}^{c} \sin{\left(a x \right)} \cos{\left(b x \right)}\, dx 
Integral(cos(b*x)*sin(a*x), (x, 0, c))
Respuesta (Indefinida) [src] 
                              //                     0                       for And(a = 0, b = 0)\
                              ||                                                                  |
                              ||                    2                                             |
                              ||                 cos (b*x)                                        |
                              ||                 ---------                        for a = -b      |
                              ||                    2*b                                           |
  /                           ||                                                                  |
 |                            ||                    2                                             |
 | cos(b*x)*sin(a*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                              |
                              \\                                                                  /
∫sin⁡(ax)cos⁡(bx) dx=C+{0for a=0∧b=0cos⁡2(bx)2bfor a=−b−cos⁡2(bx)2bfor a=b−acos⁡(ax)cos⁡(bx)a2−b2−bsin⁡(ax)sin⁡(bx)a2−b2otherwise\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}
Simplificar
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                                                                                                                                                        
|                    1    cos (b*c)                                                                                                                                                   
|                 - --- + ---------                   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*c)                                                                                                                                                    
|                  --- - ---------                                                          for Or(And(a = 0, a = b), And(a = b, b = 0), a = b)                                       
|                  2*b      2*b                                                                                                                                                       
|                                                                                                                                                                                     
|   a      a*cos(a*c)*cos(b*c)   b*sin(a*c)*sin(b*c)                                                                                                                                  
|------- - ------------------- - -------------------                                                             otherwise                                                            
| 2    2          2    2                2    2                                                                                                                                        
|a  - b          a  - b                a  - b                                                                                                                                         
\
{0for (a=0∧b=0)∨(a=0∧a=b∧b=0)∨(a=0∧a=−b∧b=0)∨(a=0∧a=−b∧a=b∧b=0)cos⁡2(bc)2b−12bfor (a=0∧a=−b)∨(a=−b∧a=b)∨(a=−b∧b=0)∨(a=0∧a=−b∧a=b)∨(a=−b∧a=b∧b=0)∨a=−b−cos⁡2(bc)2b+12bfor (a=0∧a=b)∨(a=b∧b=0)∨a=b−acos⁡(ac)cos⁡(bc)a2−b2+aa2−b2−bsin⁡(ac)sin⁡(bc)a2−b2otherwise\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 c \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 c \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 c \right)} \cos{\left(b c \right)}}{a^{2} - b^{2}} + \frac{a}{a^{2} - b^{2}} - \frac{b \sin{\left(a c \right)} \sin{\left(b c \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                                                                                                                                                        
|                    1    cos (b*c)                                                                                                                                                   
|                 - --- + ---------                   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*c)                                                                                                                                                    
|                  --- - ---------                                                          for Or(And(a = 0, a = b), And(a = b, b = 0), a = b)                                       
|                  2*b      2*b                                                                                                                                                       
|                                                                                                                                                                                     
|   a      a*cos(a*c)*cos(b*c)   b*sin(a*c)*sin(b*c)                                                                                                                                  
|------- - ------------------- - -------------------                                                             otherwise                                                            
| 2    2          2    2                2    2                                                                                                                                        
|a  - b          a  - b                a  - b                                                                                                                                         
\
{0for (a=0∧b=0)∨(a=0∧a=b∧b=0)∨(a=0∧a=−b∧b=0)∨(a=0∧a=−b∧a=b∧b=0)cos⁡2(bc)2b−12bfor (a=0∧a=−b)∨(a=−b∧a=b)∨(a=−b∧b=0)∨(a=0∧a=−b∧a=b)∨(a=−b∧a=b∧b=0)∨a=−b−cos⁡2(bc)2b+12bfor (a=0∧a=b)∨(a=b∧b=0)∨a=b−acos⁡(ac)cos⁡(bc)a2−b2+aa2−b2−bsin⁡(ac)sin⁡(bc)a2−b2otherwise\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 c \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 c \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 c \right)} \cos{\left(b c \right)}}{a^{2} - b^{2}} + \frac{a}{a^{2} - b^{2}} - \frac{b \sin{\left(a c \right)} \sin{\left(b c \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))), (-1/(2*b) + cos(b*c)^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))), (1/(2*b) - cos(b*c)^2/(2*b), (a = b)∨((a = 0)∧(a = b))∨((a = b)∧(b = 0))), (a/(a^2 - b^2) - a*cos(a*c)*cos(b*c)/(a^2 - b^2) - b*sin(a*c)*sin(b*c)/(a^2 - b^2), True))

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

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