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Resolución de integrales/ tan^4/ tan^4x/(1-tan^4x)

Integral de tan^4x/(1-tan^4x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |       4        
 |    tan (x)     
 |  ----------- dx
 |         4      
 |  1 - tan (x)   
 |                
/                 
0
∫01tan⁡4(x)1−tan⁡4(x) dx\int\limits_{0}^{1} \frac{\tan^{4}{\left(x \right)}}{1 - \tan^{4}{\left(x \right)}}\, dx 
Integral(tan(x)^4/(1 - tan(x)^4), (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

    tan⁡4(x)1−tan⁡4(x)=−tan⁡4(x)tan⁡4(x)−1\frac{\tan^{4}{\left(x \right)}}{1 - \tan^{4}{\left(x \right)}} = - \frac{\tan^{4}{\left(x \right)}}{\tan^{4}{\left(x \right)} - 1}

  2. La integral del producto de una función por una constante es la constante por la integral de esta función:

    ∫(−tan⁡4(x)tan⁡4(x)−1) dx=−∫tan⁡4(x)tan⁡4(x)−1 dx\int \left(- \frac{\tan^{4}{\left(x \right)}}{\tan^{4}{\left(x \right)} - 1}\right)\, dx = - \int \frac{\tan^{4}{\left(x \right)}}{\tan^{4}{\left(x \right)} - 1}\, dx

    1. No puedo encontrar los pasos en la búsqueda de esta integral.

      Pero la integral

      4xtan⁡2(x)8tan⁡2(x)+8+4x8tan⁡2(x)+8+log⁡(tan⁡(x)−1)tan⁡2(x)8tan⁡2(x)+8+log⁡(tan⁡(x)−1)8tan⁡2(x)+8−log⁡(tan⁡(x)+1)tan⁡2(x)8tan⁡2(x)+8−log⁡(tan⁡(x)+1)8tan⁡2(x)+8−2tan⁡(x)8tan⁡2(x)+8\frac{4 x \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{4 x}{8 \tan^{2}{\left(x \right)} + 8} + \frac{\log{\left(\tan{\left(x \right)} - 1 \right)} \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{8 \tan^{2}{\left(x \right)} + 8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{8 \tan^{2}{\left(x \right)} + 8} - \frac{2 \tan{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8}

    Por lo tanto, el resultado es: −4xtan⁡2(x)8tan⁡2(x)+8−4x8tan⁡2(x)+8−log⁡(tan⁡(x)−1)tan⁡2(x)8tan⁡2(x)+8−log⁡(tan⁡(x)−1)8tan⁡2(x)+8+log⁡(tan⁡(x)+1)tan⁡2(x)8tan⁡2(x)+8+log⁡(tan⁡(x)+1)8tan⁡2(x)+8+2tan⁡(x)8tan⁡2(x)+8- \frac{4 x \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} - \frac{4 x}{8 \tan^{2}{\left(x \right)} + 8} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)} \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{2 \tan{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8}

  3. Ahora simplificar:

    −x2−log⁡(tan⁡(x)−1)8+log⁡(tan⁡(x)+1)8+sin⁡(2x)8- \frac{x}{2} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{8} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{8} + \frac{\sin{\left(2 x \right)}}{8}

  4. Añadimos la constante de integración:

    −x2−log⁡(tan⁡(x)−1)8+log⁡(tan⁡(x)+1)8+sin⁡(2x)8+constant- \frac{x}{2} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{8} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{8} + \frac{\sin{\left(2 x \right)}}{8}+ \mathrm{constant}

Respuesta:

−x2−log⁡(tan⁡(x)−1)8+log⁡(tan⁡(x)+1)8+sin⁡(2x)8+constant- \frac{x}{2} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{8} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{8} + \frac{\sin{\left(2 x \right)}}{8}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                                                                                                                                                            
 |                                                                                                                                                             
 |      4                                                                                       2                         2                               2    
 |   tan (x)            log(1 + tan(x))   log(-1 + tan(x))        4*x           2*tan(x)     tan (x)*log(1 + tan(x))   tan (x)*log(-1 + tan(x))    4*x*tan (x) 
 | ----------- dx = C + --------------- - ---------------- - ------------- + ------------- + ----------------------- - ------------------------ - -------------
 |        4                       2                 2                 2               2                    2                         2                     2   
 | 1 - tan (x)           8 + 8*tan (x)     8 + 8*tan (x)     8 + 8*tan (x)   8 + 8*tan (x)        8 + 8*tan (x)             8 + 8*tan (x)         8 + 8*tan (x)
 |                                                                                                                                                             
/
∫tan⁡4(x)1−tan⁡4(x) dx=C−4xtan⁡2(x)8tan⁡2(x)+8−4x8tan⁡2(x)+8−log⁡(tan⁡(x)−1)tan⁡2(x)8tan⁡2(x)+8−log⁡(tan⁡(x)−1)8tan⁡2(x)+8+log⁡(tan⁡(x)+1)tan⁡2(x)8tan⁡2(x)+8+log⁡(tan⁡(x)+1)8tan⁡2(x)+8+2tan⁡(x)8tan⁡2(x)+8\int \frac{\tan^{4}{\left(x \right)}}{1 - \tan^{4}{\left(x \right)}}\, dx = C - \frac{4 x \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} - \frac{4 x}{8 \tan^{2}{\left(x \right)} + 8} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)} \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{2 \tan{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8}
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Gráfica
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Trazar el gráfico f(x)
Respuesta [src] 
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Respuesta numérica [src] 
-0.545789771331317
-0.545789771331317

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

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