Integral de tanx*tan(x+a) dx

Solución

Ha introducido [src]

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$$\int\limits_{0}^{0} \tan{\left(x \right)} \tan{\left(a + x \right)}\, dx$$

Integral(tan(x)*tan(x + a), (x, 0, 0))

Solución detallada

Vuelva a escribir el integrando:

$\tan{\left(x \right)} \tan{\left(a + x \right)} = \frac{\tan{\left(a \right)} \tan{\left(x \right)}}{- \tan{\left(a \right)} \tan{\left(x \right)} + 1} + \frac{\tan^{2}{\left(x \right)}}{- \tan{\left(a \right)} \tan{\left(x \right)} + 1}$
Integramos término a término:
1. La integral del producto de una función por una constante es la constante por la integral de esta función:
  
  $\int \frac{\tan{\left(a \right)} \tan{\left(x \right)}}{- \tan{\left(a \right)} \tan{\left(x \right)} + 1}\, dx = \tan{\left(a \right)} \int \frac{\tan{\left(x \right)}}{- \tan{\left(a \right)} \tan{\left(x \right)} + 1}\, dx$
  1. Vuelva a escribir el integrando:
    
    $\frac{\tan{\left(x \right)}}{- \tan{\left(a \right)} \tan{\left(x \right)} + 1} = - \frac{\tan{\left(x \right)}}{\tan{\left(a \right)} \tan{\left(x \right)} - 1}$
  2. La integral del producto de una función por una constante es la constante por la integral de esta función:
    
    $\int \left(- \frac{\tan{\left(x \right)}}{\tan{\left(a \right)} \tan{\left(x \right)} - 1}\right)\, dx = - \int \frac{\tan{\left(x \right)}}{\tan{\left(a \right)} \tan{\left(x \right)} - 1}\, dx$
    1. No puedo encontrar los pasos en la búsqueda de esta integral.
      
      Pero la integral
      
      $\begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{2}{\left(a \right)} + 2} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{2}{\left(a \right)} + 2} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2 \tan^{2}{\left(a \right)} + 2} & \text{for}\: a \neq 0 \\- \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2} & \text{otherwese} \end{cases}$
    Por lo tanto, el resultado es: $- \begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{2}{\left(a \right)} + 2} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{2}{\left(a \right)} + 2} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2 \tan^{2}{\left(a \right)} + 2} & \text{for}\: a \neq 0 \\- \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2} & \text{otherwese} \end{cases}$
  Por lo tanto, el resultado es: $- \left(\begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{2}{\left(a \right)} + 2} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{2}{\left(a \right)} + 2} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2 \tan^{2}{\left(a \right)} + 2} & \text{for}\: a \neq 0 \\- \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2} & \text{otherwese} \end{cases}\right) \tan{\left(a \right)}$
1. Vuelva a escribir el integrando:
  
  $\frac{\tan^{2}{\left(x \right)}}{- \tan{\left(a \right)} \tan{\left(x \right)} + 1} = - \frac{\tan^{2}{\left(x \right)}}{\tan{\left(a \right)} \tan{\left(x \right)} - 1}$
2. La integral del producto de una función por una constante es la constante por la integral de esta función:
  
  $\int \left(- \frac{\tan^{2}{\left(x \right)}}{\tan{\left(a \right)} \tan{\left(x \right)} - 1}\right)\, dx = - \int \frac{\tan^{2}{\left(x \right)}}{\tan{\left(a \right)} \tan{\left(x \right)} - 1}\, dx$
  1. No puedo encontrar los pasos en la búsqueda de esta integral.
    
    Pero la integral
    
    $\begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} & \text{for}\: a \neq 0 \\x - \tan{\left(x \right)} & \text{otherwese} \end{cases}$
  Por lo tanto, el resultado es: $- \begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} & \text{for}\: a \neq 0 \\x - \tan{\left(x \right)} & \text{otherwese} \end{cases}$
El resultado es: $- \left(\begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{2}{\left(a \right)} + 2} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{2}{\left(a \right)} + 2} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2 \tan^{2}{\left(a \right)} + 2} & \text{for}\: a \neq 0 \\- \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2} & \text{otherwese} \end{cases}\right) \tan{\left(a \right)} - \begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} & \text{for}\: a \neq 0 \\x - \tan{\left(x \right)} & \text{otherwese} \end{cases}$
Ahora simplificar:

$\begin{cases} - (x + \frac{\log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{\tan{\left(a \right)}}) & \text{for}\: a \neq 0 \\- x + \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)} \tan{\left(a \right)}}{2} + \tan{\left(x \right)} & \text{otherwese} \end{cases}$
Añadimos la constante de integración:

$\begin{cases} - (x + \frac{\log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{\tan{\left(a \right)}}) & \text{for}\: a \neq 0 \\- x + \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)} \tan{\left(a \right)}}{2} + \tan{\left(x \right)} & \text{otherwese} \end{cases}+ \mathrm{constant}$

Respuesta:

$\begin{cases} - (x + \frac{\log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{\tan{\left(a \right)}}) & \text{for}\: a \neq 0 \\- x + \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)} \tan{\left(a \right)}}{2} + \tan{\left(x \right)} & \text{otherwese} \end{cases}+ \mathrm{constant}$

Respuesta (Indefinida) [src]

                                                                                                                          //                          /    1            \                            \       
                              //     /    1            \                                                              \   ||     /       2   \   2*log|- ------ + tan(x)|                            |       
                              ||2*log|- ------ + tan(x)|      2       /       2   \                                   |   ||  log\1 + tan (x)/        \  tan(a)         /     2*x*tan(a)             |       
  /                           ||     \  tan(a)         /   tan (a)*log\1 + tan (x)/        2*x*tan(a)                 |   ||- ---------------- + ------------------------ + -------------  for a != 0|       
 |                            ||------------------------ + ------------------------ + --------------------  for a != 0|   ||            2                      2                     2               |       
 | tan(x)*tan(x + a) dx = C - |<       3                          3                        3                          | - |<   2 + 2*tan (a)          2 + 2*tan (a)         2 + 2*tan (a)            |*tan(a)
 |                            ||  2*tan (a) + 2*tan(a)       2*tan (a) + 2*tan(a)     2*tan (a) + 2*tan(a)            |   ||                                                                         |       
/                             ||                                                                                      |   ||                         /       2   \                                   |       
                              ||                                x - tan(x)                                  otherwise |   ||                     -log\1 + tan (x)/                                   |       
                              \\                                                                                      /   ||                     ------------------                        otherwise |       
                                                                                                                          \\                             2                                           /

$$\int \tan{\left(x \right)} \tan{\left(a + x \right)}\, dx = C - \left(\begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{2}{\left(a \right)} + 2} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{2}{\left(a \right)} + 2} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2 \tan^{2}{\left(a \right)} + 2} & \text{for}\: a \neq 0 \\- \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2} & \text{otherwise} \end{cases}\right) \tan{\left(a \right)} - \begin{cases} \frac{2 x \tan{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{2 \log{\left(\tan{\left(x \right)} - \frac{1}{\tan{\left(a \right)}} \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(a \right)}}{2 \tan^{3}{\left(a \right)} + 2 \tan{\left(a \right)}} & \text{for}\: a \neq 0 \\x - \tan{\left(x \right)} & \text{otherwise} \end{cases}$$

Simplificar

Respuesta [src]

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Respuesta numérica [src]

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Integral de tanx*tan(x+a) dx

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Solución