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Resolución de integrales/ cosbx/ cosax/ cosax×cosbx

Integral de cosax×cosbx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                     
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 |  cos(a*x)*cos(b*x) dx
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0
∫01cos⁡(ax)cos⁡(bx) dx\int\limits_{0}^{1} \cos{\left(a x \right)} \cos{\left(b x \right)}\, dx 
Integral(cos(a*x)*cos(b*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
                              //                      x                        for And(a = 0, b = 0)\
                              ||                                                                    |
                              ||     2             2                                                |
  /                           ||x*cos (b*x)   x*sin (b*x)   cos(b*x)*sin(b*x)                       |
 |                            ||----------- + ----------- + -----------------  for Or(a = -b, a = b)|
 | cos(a*x)*cos(b*x) dx = C + |<     2             2               2*b                              |
 |                            ||                                                                    |
/                             ||  a*cos(b*x)*sin(a*x)   b*cos(a*x)*sin(b*x)                         |
                              ||  ------------------- - -------------------          otherwise      |
                              ||         2    2                2    2                               |
                              \\        a  - b                a  - b                                /
∫cos⁡(ax)cos⁡(bx) dx=C+{xfor a=0∧b=0xsin⁡2(bx)2+xcos⁡2(bx)2+sin⁡(bx)cos⁡(bx)2bfor a=−b∨a=basin⁡(ax)cos⁡(bx)a2−b2−bsin⁡(bx)cos⁡(ax)a2−b2otherwise\int \cos{\left(a x \right)} \cos{\left(b x \right)}\, dx = C + \begin{cases} x & \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 \vee a = b \\\frac{a \sin{\left(a x \right)} \cos{\left(b x \right)}}{a^{2} - b^{2}} - \frac{b \sin{\left(b x \right)} \cos{\left(a x \right)}}{a^{2} - b^{2}} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/                1                                                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 = 0, a = b), And(a = -b, a = b), And(a = b, b = 0), And(a = -b, b = 0), And(a = 0, a = -b, a = b), And(a = -b, a = b, b = 0), a = -b, a = b)
<   2         2           2*b                                                                                                                                                                                    
|                                                                                                                                                                                                                
|a*cos(b)*sin(a)   b*cos(a)*sin(b)                                                                                                                                                                               
|--------------- - ---------------                                                                                    otherwise                                                                                  
|     2    2            2    2                                                                                                                                                                                   
\    a  - b            a  - b
{1for (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=0∧a=b)∨(a=−b∧a=b)∨(a=b∧b=0)∨(a=−b∧b=0)∨(a=0∧a=−b∧a=b)∨(a=−b∧a=b∧b=0)∨a=−b∨a=basin⁡(a)cos⁡(b)a2−b2−bsin⁡(b)cos⁡(a)a2−b2otherwise\begin{cases} 1 & \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 = 0 \wedge a = b\right) \vee \left(a = - b \wedge a = b\right) \vee \left(a = b \wedge b = 0\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 \vee a = b \\\frac{a \sin{\left(a \right)} \cos{\left(b \right)}}{a^{2} - b^{2}} - \frac{b \sin{\left(b \right)} \cos{\left(a \right)}}{a^{2} - b^{2}} & \text{otherwise} \end{cases} 
=
= 
/                1                                                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 = 0, a = b), And(a = -b, a = b), And(a = b, b = 0), And(a = -b, b = 0), And(a = 0, a = -b, a = b), And(a = -b, a = b, b = 0), a = -b, a = b)
<   2         2           2*b                                                                                                                                                                                    
|                                                                                                                                                                                                                
|a*cos(b)*sin(a)   b*cos(a)*sin(b)                                                                                                                                                                               
|--------------- - ---------------                                                                                    otherwise                                                                                  
|     2    2            2    2                                                                                                                                                                                   
\    a  - b            a  - b
{1for (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=0∧a=b)∨(a=−b∧a=b)∨(a=b∧b=0)∨(a=−b∧b=0)∨(a=0∧a=−b∧a=b)∨(a=−b∧a=b∧b=0)∨a=−b∨a=basin⁡(a)cos⁡(b)a2−b2−bsin⁡(b)cos⁡(a)a2−b2otherwise\begin{cases} 1 & \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 = 0 \wedge a = b\right) \vee \left(a = - b \wedge a = b\right) \vee \left(a = b \wedge b = 0\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 \vee a = b \\\frac{a \sin{\left(a \right)} \cos{\left(b \right)}}{a^{2} - b^{2}} - \frac{b \sin{\left(b \right)} \cos{\left(a \right)}}{a^{2} - b^{2}} & \text{otherwise} \end{cases} 
Piecewise((1, ((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 = -b)∨((a = 0)∧(a = b))∨((a = b)∧(b = 0))∨((a = 0)∧(a = -b))∨((a = b)∧(a = -b))∨((b = 0)∧(a = -b))∨((a = 0)∧(a = b)∧(a = -b))∨((a = b)∧(b = 0)∧(a = -b))), (a*cos(b)*sin(a)/(a^2 - b^2) - b*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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