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Resolución de integrales/ cos(x)/ cos(ax)/ cos(x)×cos(ax)

Integral de cos(x)×cos(ax) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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

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