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Resolución de integrales/ tan^3/ tan^2/ (1+tan^2x)/(1+tan^3x)

Integral de (1+tan^2x)/(1+tan^3x) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1               
  /               
 |                
 |         2      
 |  1 + tan (x)   
 |  ----------- dx
 |         3      
 |  1 + tan (x)   
 |                
/                 
0
∫01tan⁡2(x)+1tan⁡3(x)+1 dx\int\limits_{0}^{1} \frac{\tan^{2}{\left(x \right)} + 1}{\tan^{3}{\left(x \right)} + 1}\, dx 
Integral((1 + tan(x)^2)/(1 + tan(x)^3), (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

    tan⁡2(x)+1tan⁡3(x)+1=tan⁡2(x)tan⁡3(x)+1+1tan⁡3(x)+1\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{3}{\left(x \right)} + 1} = \frac{\tan^{2}{\left(x \right)}}{\tan^{3}{\left(x \right)} + 1} + \frac{1}{\tan^{3}{\left(x \right)} + 1}

  2. Integramos término a término:

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

      Pero la integral

      −x2+3(atan⁡(23tan⁡(x)3−33)+π⌊x−π2π⌋)3+log⁡(tan⁡(x)+1)6−log⁡(tan⁡2(x)+1)4+log⁡(4tan⁡2(x)−4tan⁡(x)+4)6- \frac{x}{2} + \frac{\sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(x \right)}}{3} - \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{x - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{6} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(4 \tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)} + 4 \right)}}{6}

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

      Pero la integral

      x2+log⁡(tan⁡(x)+1)6+log⁡(tan⁡2(x)+1)4−log⁡(tan⁡2(x)−tan⁡(x)+1)3\frac{x}{2} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{6} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4} - \frac{\log{\left(\tan^{2}{\left(x \right)} - \tan{\left(x \right)} + 1 \right)}}{3}

    El resultado es: 3(atan⁡(23tan⁡(x)3−33)+π⌊x−π2π⌋)3+log⁡(tan⁡(x)+1)3−log⁡(tan⁡2(x)−tan⁡(x)+1)3+log⁡(4tan⁡2(x)−4tan⁡(x)+4)6\frac{\sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(x \right)}}{3} - \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{x - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{3} - \frac{\log{\left(\tan^{2}{\left(x \right)} - \tan{\left(x \right)} + 1 \right)}}{3} + \frac{\log{\left(4 \tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)} + 4 \right)}}{6}

  3. Ahora simplificar:

    −log⁡(−tan⁡(x)+1cos⁡2(x))6+log⁡(tan⁡(x)+1)3+3atan⁡(3(2tan⁡(x)−1)3)3+3π⌊xπ−12⌋3+log⁡(2)3- \frac{\log{\left(- \tan{\left(x \right)} + \frac{1}{\cos^{2}{\left(x \right)}} \right)}}{6} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{3} + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(2 \tan{\left(x \right)} - 1\right)}{3} \right)}}{3} + \frac{\sqrt{3} \pi \left\lfloor{\frac{x}{\pi} - \frac{1}{2}}\right\rfloor}{3} + \frac{\log{\left(2 \right)}}{3}

  4. Añadimos la constante de integración:

    −log⁡(−tan⁡(x)+1cos⁡2(x))6+log⁡(tan⁡(x)+1)3+3atan⁡(3(2tan⁡(x)−1)3)3+3π⌊xπ−12⌋3+log⁡(2)3+constant- \frac{\log{\left(- \tan{\left(x \right)} + \frac{1}{\cos^{2}{\left(x \right)}} \right)}}{6} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{3} + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(2 \tan{\left(x \right)} - 1\right)}{3} \right)}}{3} + \frac{\sqrt{3} \pi \left\lfloor{\frac{x}{\pi} - \frac{1}{2}}\right\rfloor}{3} + \frac{\log{\left(2 \right)}}{3}+ \mathrm{constant}

Respuesta:

−log⁡(−tan⁡(x)+1cos⁡2(x))6+log⁡(tan⁡(x)+1)3+3atan⁡(3(2tan⁡(x)−1)3)3+3π⌊xπ−12⌋3+log⁡(2)3+constant- \frac{\log{\left(- \tan{\left(x \right)} + \frac{1}{\cos^{2}{\left(x \right)}} \right)}}{6} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{3} + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(2 \tan{\left(x \right)} - 1\right)}{3} \right)}}{3} + \frac{\sqrt{3} \pi \left\lfloor{\frac{x}{\pi} - \frac{1}{2}}\right\rfloor}{3} + \frac{\log{\left(2 \right)}}{3}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                                                                                                            /        /    pi\                                 \
  /                                                                                                         |        |x - --|       /    ___       ___       \|
 |                                                                                                      ___ |        |    2 |       |  \/ 3    2*\/ 3 *tan(x)||
 |        2                /       2            \                        /                    2   \   \/ 3 *|pi*floor|------| + atan|- ----- + --------------||
 | 1 + tan (x)          log\1 + tan (x) - tan(x)/   log(1 + tan(x))   log\4 - 4*tan(x) + 4*tan (x)/         \        \  pi  /       \    3           3       //
 | ----------- dx = C - ------------------------- + --------------- + ----------------------------- + ---------------------------------------------------------
 |        3                         3                      3                        6                                             3                            
 | 1 + tan (x)                                                                                                                                                 
 |                                                                                                                                                             
/
∫tan⁡2(x)+1tan⁡3(x)+1 dx=C+3(atan⁡(23tan⁡(x)3−33)+π⌊x−π2π⌋)3+log⁡(tan⁡(x)+1)3−log⁡(tan⁡2(x)−tan⁡(x)+1)3+log⁡(4tan⁡2(x)−4tan⁡(x)+4)6\int \frac{\tan^{2}{\left(x \right)} + 1}{\tan^{3}{\left(x \right)} + 1}\, dx = C + \frac{\sqrt{3} \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \tan{\left(x \right)}}{3} - \frac{\sqrt{3}}{3} \right)} + \pi \left\lfloor{\frac{x - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{3} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{3} - \frac{\log{\left(\tan^{2}{\left(x \right)} - \tan{\left(x \right)} + 1 \right)}}{3} + \frac{\log{\left(4 \tan^{2}{\left(x \right)} - 4 \tan{\left(x \right)} + 4 \right)}}{6}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.905-5
Trazar el gráfico f(x)
Respuesta [src] 
                                                                   /          /  ___       ___       \\             
                                                               ___ |          |\/ 3    2*\/ 3 *tan(1)||             
     /                    2   \                              \/ 3 *|-pi - atan|----- - --------------||          ___
  log\4 - 4*tan(1) + 4*tan (1)/   log(1 + tan(1))   log(4)         \          \  3           3       //   7*pi*\/ 3 
- ----------------------------- + --------------- + ------ + ------------------------------------------ + ----------
                6                        3            6                          3                            18
3(−π−atan⁡(−23tan⁡(1)3+33))3−log⁡(−4tan⁡(1)+4+4tan⁡2(1))6+log⁡(4)6+log⁡(1+tan⁡(1))3+73π18\frac{\sqrt{3} \left(- \pi - \operatorname{atan}{\left(- \frac{2 \sqrt{3} \tan{\left(1 \right)}}{3} + \frac{\sqrt{3}}{3} \right)}\right)}{3} - \frac{\log{\left(- 4 \tan{\left(1 \right)} + 4 + 4 \tan^{2}{\left(1 \right)} \right)}}{6} + \frac{\log{\left(4 \right)}}{6} + \frac{\log{\left(1 + \tan{\left(1 \right)} \right)}}{3} + \frac{7 \sqrt{3} \pi}{18} 
=
= 
                                                                   /          /  ___       ___       \\             
                                                               ___ |          |\/ 3    2*\/ 3 *tan(1)||             
     /                    2   \                              \/ 3 *|-pi - atan|----- - --------------||          ___
  log\4 - 4*tan(1) + 4*tan (1)/   log(1 + tan(1))   log(4)         \          \  3           3       //   7*pi*\/ 3 
- ----------------------------- + --------------- + ------ + ------------------------------------------ + ----------
                6                        3            6                          3                            18
3(−π−atan⁡(−23tan⁡(1)3+33))3−log⁡(−4tan⁡(1)+4+4tan⁡2(1))6+log⁡(4)6+log⁡(1+tan⁡(1))3+73π18\frac{\sqrt{3} \left(- \pi - \operatorname{atan}{\left(- \frac{2 \sqrt{3} \tan{\left(1 \right)}}{3} + \frac{\sqrt{3}}{3} \right)}\right)}{3} - \frac{\log{\left(- 4 \tan{\left(1 \right)} + 4 + 4 \tan^{2}{\left(1 \right)} \right)}}{6} + \frac{\log{\left(4 \right)}}{6} + \frac{\log{\left(1 + \tan{\left(1 \right)} \right)}}{3} + \frac{7 \sqrt{3} \pi}{18} 
-log(4 - 4*tan(1) + 4*tan(1)^2)/6 + log(1 + tan(1))/3 + log(4)/6 + sqrt(3)*(-pi - atan(sqrt(3)/3 - 2*sqrt(3)*tan(1)/3))/3 + 7*pi*sqrt(3)/18
Respuesta numérica [src] 
1.02185125345814
1.02185125345814

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

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