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Resolución de integrales/ (x/2)^2/ tan(x/2)/ tan(x/2)/(1-tan(x/2)^2)

Integral de tan(x/2)/(1-tan(x/2)^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |        /x\     
 |     tan|-|     
 |        \2/     
 |  ----------- dx
 |         2/x\   
 |  1 - tan |-|   
 |          \2/   
 |                
/                 
0
∫01tan⁡(x2)1−tan⁡2(x2) dx\int\limits_{0}^{1} \frac{\tan{\left(\frac{x}{2} \right)}}{1 - \tan^{2}{\left(\frac{x}{2} \right)}}\, dx 
Integral(tan(x/2)/(1 - tan(x/2)^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                       //      /   2/x\\                  \
 |                        ||-acoth|tan |-||                  |
 |       /x\              ||      \    \2//          4/x\    |
 |    tan|-|              ||----------------  for tan |-| > 1|
 |       \2/              ||       2                  \2/    |
 | ----------- dx = C - 2*|<                                 |
 |        2/x\            ||      /   2/x\\                  |
 | 1 - tan |-|            ||-atanh|tan |-||                  |
 |         \2/            ||      \    \2//          4/x\    |
 |                        ||----------------  for tan |-| < 1|
/                         \\       2                  \2/    /
∫tan⁡(x2)1−tan⁡2(x2) dx=C−2({−acoth⁡(tan⁡2(x2))2for tan⁡4(x2)>1−atanh⁡(tan⁡2(x2))2for tan⁡4(x2)<1)\int \frac{\tan{\left(\frac{x}{2} \right)}}{1 - \tan^{2}{\left(\frac{x}{2} \right)}}\, dx = C - 2 \left(\begin{cases} - \frac{\operatorname{acoth}{\left(\tan^{2}{\left(\frac{x}{2} \right)} \right)}}{2} & \text{for}\: \tan^{4}{\left(\frac{x}{2} \right)} > 1 \\- \frac{\operatorname{atanh}{\left(\tan^{2}{\left(\frac{x}{2} \right)} \right)}}{2} & \text{for}\: \tan^{4}{\left(\frac{x}{2} \right)} < 1 \end{cases}\right)
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.01.0
Trazar el gráfico f(x)
Respuesta [src] 
   /       2     \                                        
log\1 + tan (1/2)/   log(1 - tan(1/2))   log(1 + tan(1/2))
------------------ - ----------------- - -----------------
        2                    2                   2
−log⁡(tan⁡(12)+1)2+log⁡(tan⁡2(12)+1)2−log⁡(1−tan⁡(12))2- \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{2} + \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{2} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)}}{2} 
=
= 
   /       2     \                                        
log\1 + tan (1/2)/   log(1 - tan(1/2))   log(1 + tan(1/2))
------------------ - ----------------- - -----------------
        2                    2                   2
−log⁡(tan⁡(12)+1)2+log⁡(tan⁡2(12)+1)2−log⁡(1−tan⁡(12))2- \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{2} + \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{2} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)}}{2} 
log(1 + tan(1/2)^2)/2 - log(1 - tan(1/2))/2 - log(1 + tan(1/2))/2
Respuesta numérica [src] 
0.307813235193007
0.307813235193007

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

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