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Resolución de integrales/ sinax/ sinax*sinbx

Integral de sinax*sinbx 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)*sin(b*x) dx
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0
∫01sin⁡(ax)sin⁡(bx) dx\int\limits_{0}^{1} \sin{\left(a x \right)} \sin{\left(b x \right)}\, dx 
Integral(sin(a*x)*sin(b*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                              //                       0                         for And(a = 0, b = 0)\
                              ||                                                                      |
                              ||       2             2                                                |
                              ||  x*cos (b*x)   x*sin (b*x)   cos(b*x)*sin(b*x)                       |
                              ||- ----------- - ----------- + -----------------       for a = -b      |
                              ||       2             2               2*b                              |
  /                           ||                                                                      |
 |                            ||      2             2                                                 |
 | sin(a*x)*sin(b*x) dx = C + |< x*cos (b*x)   x*sin (b*x)   cos(b*x)*sin(b*x)                        |
 |                            || ----------- + ----------- - -----------------         for a = b      |
/                             ||      2             2               2*b                               |
                              ||                                                                      |
                              ||   b*cos(b*x)*sin(a*x)   a*cos(a*x)*sin(b*x)                          |
                              ||   ------------------- - -------------------           otherwise      |
                              ||          2    2                2    2                                |
                              ||         a  - b                a  - b                                 |
                              \\                                                                      /
∫sin⁡(ax)sin⁡(bx) dx=C+{0for a=0∧b=0−xsin⁡2(bx)2−xcos⁡2(bx)2+sin⁡(bx)cos⁡(bx)2bfor a=−bxsin⁡2(bx)2+xcos⁡2(bx)2−sin⁡(bx)cos⁡(bx)2bfor a=b−asin⁡(bx)cos⁡(ax)a2−b2+bsin⁡(ax)cos⁡(bx)a2−b2otherwise\int \sin{\left(a x \right)} \sin{\left(b x \right)}\, dx = C + \begin{cases} 0 & \text{for}\: a = 0 \wedge b = 0 \\- \frac{x \sin^{2}{\left(b x \right)}}{2} - \frac{x \cos^{2}{\left(b x \right)}}{2} + \frac{\sin{\left(b x \right)} \cos{\left(b x \right)}}{2 b} & \text{for}\: a = - b \\\frac{x \sin^{2}{\left(b x \right)}}{2} + \frac{x \cos^{2}{\left(b x \right)}}{2} - \frac{\sin{\left(b x \right)} \cos{\left(b x \right)}}{2 b} & \text{for}\: a = b \\- \frac{a \sin{\left(b x \right)} \cos{\left(a x \right)}}{a^{2} - b^{2}} + \frac{b \sin{\left(a x \right)} \cos{\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         2                                                                                                                                                     
|  cos (b)   sin (b)   cos(b)*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         2           2*b                                                                                                                                       
|                                                                                                                                                                     
|    2         2                                                                                                                                                      
< cos (b)   sin (b)   cos(b)*sin(b)                                                                                                                                   
| ------- + ------- - -------------                                         for Or(And(a = 0, a = b), And(a = b, b = 0), a = b)                                       
|    2         2           2*b                                                                                                                                        
|                                                                                                                                                                     
| b*cos(b)*sin(a)   a*cos(a)*sin(b)                                                                                                                                   
| --------------- - ---------------                                                              otherwise                                                            
|      2    2            2    2                                                                                                                                       
|     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)−sin⁡2(b)2−cos⁡2(b)2+sin⁡(b)cos⁡(b)2bfor (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=−bsin⁡2(b)2+cos⁡2(b)2−sin⁡(b)cos⁡(b)2bfor (a=0∧a=b)∨(a=b∧b=0)∨a=b−asin⁡(b)cos⁡(a)a2−b2+bsin⁡(a)cos⁡(b)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{\sin^{2}{\left(b \right)}}{2} - \frac{\cos^{2}{\left(b \right)}}{2} + \frac{\sin{\left(b \right)} \cos{\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} + \frac{\cos^{2}{\left(b \right)}}{2} - \frac{\sin{\left(b \right)} \cos{\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 \sin{\left(b \right)} \cos{\left(a \right)}}{a^{2} - b^{2}} + \frac{b \sin{\left(a \right)} \cos{\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         2                                                                                                                                                     
|  cos (b)   sin (b)   cos(b)*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         2           2*b                                                                                                                                       
|                                                                                                                                                                     
|    2         2                                                                                                                                                      
< cos (b)   sin (b)   cos(b)*sin(b)                                                                                                                                   
| ------- + ------- - -------------                                         for Or(And(a = 0, a = b), And(a = b, b = 0), a = b)                                       
|    2         2           2*b                                                                                                                                        
|                                                                                                                                                                     
| b*cos(b)*sin(a)   a*cos(a)*sin(b)                                                                                                                                   
| --------------- - ---------------                                                              otherwise                                                            
|      2    2            2    2                                                                                                                                       
|     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)−sin⁡2(b)2−cos⁡2(b)2+sin⁡(b)cos⁡(b)2bfor (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=−bsin⁡2(b)2+cos⁡2(b)2−sin⁡(b)cos⁡(b)2bfor (a=0∧a=b)∨(a=b∧b=0)∨a=b−asin⁡(b)cos⁡(a)a2−b2+bsin⁡(a)cos⁡(b)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{\sin^{2}{\left(b \right)}}{2} - \frac{\cos^{2}{\left(b \right)}}{2} + \frac{\sin{\left(b \right)} \cos{\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} + \frac{\cos^{2}{\left(b \right)}}{2} - \frac{\sin{\left(b \right)} \cos{\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 \sin{\left(b \right)} \cos{\left(a \right)}}{a^{2} - b^{2}} + \frac{b \sin{\left(a \right)} \cos{\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))), (-cos(b)^2/2 - sin(b)^2/2 + cos(b)*sin(b)/(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))), (cos(b)^2/2 + sin(b)^2/2 - cos(b)*sin(b)/(2*b), (a = b)∨((a = 0)∧(a = b))∨((a = b)∧(b = 0))), (b*cos(b)*sin(a)/(a^2 - b^2) - a*cos(a)*sin(b)/(a^2 - b^2), True))

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

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