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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
      2                     
      /                     
     |                      
     |         /   1    \   
     |     atan|--------|   
     |         |       2|   
     |         \(x + 1) /   
     |     -------------- dx
     |               3      
     |        (x + 1)       
     |                      
    /                       
       ___                  
-1 + \/ 3
∫−1+32atan⁡(1(x+1)2)(x+1)3 dx\int\limits_{-1 + \sqrt{3}}^{2} \frac{\operatorname{atan}{\left(\frac{1}{\left(x + 1\right)^{2}} \right)}}{\left(x + 1\right)^{3}}\, dx 
Integral(atan(1/((x + 1)^2))/(x + 1)^3, (x, -1 + sqrt(3), 2))
Gráfica
2.00.80.91.01.11.21.31.41.51.61.71.81.90.10-0.10
Trazar el gráfico f(x)
Respuesta [src] 
                                                                                                              /             1              \                                                                                                                                                                                                                                                                                
                                                                                                        2*atan|----------------------------|                                                                                                                                                                                                                                                                                
                                   /                 4                           3                 2\         |                 2          |                                                2    /                 4                           3                 2\                     /                 4                           3                 2\                   2                                              
                                   |     /       ___\        ___     /       ___\      /       ___\ |         |     /       ___\        ___|                 /  ___\            /       ___\     |     /       ___\        ___     /       ___\      /       ___\ |     /       ___\    |     /       ___\        ___     /       ___\      /       ___\ |       /       ___\     /  ___\         /       ___\    /  ___\   
          atan(1/9)   log(82)   log\-2 + \-1 + \/ 3 /  + 4*\/ 3  + 4*\-1 + \/ 3 /  + 6*\-1 + \/ 3 / /         \-1 + \-1 + \/ 3 /  + 2*\/ 3 /            4*log\\/ 3 /            \-1 + \/ 3 / *log\-2 + \-1 + \/ 3 /  + 4*\/ 3  + 4*\-1 + \/ 3 /  + 6*\-1 + \/ 3 / /   2*\-1 + \/ 3 /*log\-2 + \-1 + \/ 3 /  + 4*\/ 3  + 4*\-1 + \/ 3 /  + 6*\-1 + \/ 3 / /     4*\-1 + \/ 3 / *log\\/ 3 /       8*\-1 + \/ 3 /*log\\/ 3 /   
-log(3) - --------- + ------- - --------------------------------------------------------------------- + ------------------------------------ + ------------------------------ - ----------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------ + ------------------------------ + ------------------------------
              18         4                                             2                                                      2                                   2                                                           2                                                                                     2                                                           2                                2          
                                                           /       ___\        ___                                /       ___\        ___             /       ___\        ___                                     /       ___\        ___                                                               /       ___\        ___                                     /       ___\        ___          /       ___\        ___
                                                    -4 + 4*\-1 + \/ 3 /  + 8*\/ 3                          -4 + 4*\-1 + \/ 3 /  + 8*\/ 3       -4 + 4*\-1 + \/ 3 /  + 8*\/ 3                               -4 + 4*\-1 + \/ 3 /  + 8*\/ 3                                                         -4 + 4*\-1 + \/ 3 /  + 8*\/ 3                               -4 + 4*\-1 + \/ 3 /  + 8*\/ 3    -4 + 4*\-1 + \/ 3 /  + 8*\/ 3
−log⁡(3)−2(−1+3)log⁡(−2+(−1+3)4+4(−1+3)3+6(−1+3)2+43)−4+4(−1+3)2+83−log⁡(−2+(−1+3)4+4(−1+3)3+6(−1+3)2+43)−4+4(−1+3)2+83−(−1+3)2log⁡(−2+(−1+3)4+4(−1+3)3+6(−1+3)2+43)−4+4(−1+3)2+83−atan⁡(19)18+2atan⁡(1−1+(−1+3)2+23)−4+4(−1+3)2+83+4(−1+3)2log⁡(3)−4+4(−1+3)2+83+4log⁡(3)−4+4(−1+3)2+83+8(−1+3)log⁡(3)−4+4(−1+3)2+83+log⁡(82)4- \log{\left(3 \right)} - \frac{2 \left(-1 + \sqrt{3}\right) \log{\left(-2 + \left(-1 + \sqrt{3}\right)^{4} + 4 \left(-1 + \sqrt{3}\right)^{3} + 6 \left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} - \frac{\log{\left(-2 + \left(-1 + \sqrt{3}\right)^{4} + 4 \left(-1 + \sqrt{3}\right)^{3} + 6 \left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} - \frac{\left(-1 + \sqrt{3}\right)^{2} \log{\left(-2 + \left(-1 + \sqrt{3}\right)^{4} + 4 \left(-1 + \sqrt{3}\right)^{3} + 6 \left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} - \frac{\operatorname{atan}{\left(\frac{1}{9} \right)}}{18} + \frac{2 \operatorname{atan}{\left(\frac{1}{-1 + \left(-1 + \sqrt{3}\right)^{2} + 2 \sqrt{3}} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{4 \left(-1 + \sqrt{3}\right)^{2} \log{\left(\sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{4 \log{\left(\sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{8 \left(-1 + \sqrt{3}\right) \log{\left(\sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{\log{\left(82 \right)}}{4} 
=
= 
                                                                                                              /             1              \                                                                                                                                                                                                                                                                                
                                                                                                        2*atan|----------------------------|                                                                                                                                                                                                                                                                                
                                   /                 4                           3                 2\         |                 2          |                                                2    /                 4                           3                 2\                     /                 4                           3                 2\                   2                                              
                                   |     /       ___\        ___     /       ___\      /       ___\ |         |     /       ___\        ___|                 /  ___\            /       ___\     |     /       ___\        ___     /       ___\      /       ___\ |     /       ___\    |     /       ___\        ___     /       ___\      /       ___\ |       /       ___\     /  ___\         /       ___\    /  ___\   
          atan(1/9)   log(82)   log\-2 + \-1 + \/ 3 /  + 4*\/ 3  + 4*\-1 + \/ 3 /  + 6*\-1 + \/ 3 / /         \-1 + \-1 + \/ 3 /  + 2*\/ 3 /            4*log\\/ 3 /            \-1 + \/ 3 / *log\-2 + \-1 + \/ 3 /  + 4*\/ 3  + 4*\-1 + \/ 3 /  + 6*\-1 + \/ 3 / /   2*\-1 + \/ 3 /*log\-2 + \-1 + \/ 3 /  + 4*\/ 3  + 4*\-1 + \/ 3 /  + 6*\-1 + \/ 3 / /     4*\-1 + \/ 3 / *log\\/ 3 /       8*\-1 + \/ 3 /*log\\/ 3 /   
-log(3) - --------- + ------- - --------------------------------------------------------------------- + ------------------------------------ + ------------------------------ - ----------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------ + ------------------------------ + ------------------------------
              18         4                                             2                                                      2                                   2                                                           2                                                                                     2                                                           2                                2          
                                                           /       ___\        ___                                /       ___\        ___             /       ___\        ___                                     /       ___\        ___                                                               /       ___\        ___                                     /       ___\        ___          /       ___\        ___
                                                    -4 + 4*\-1 + \/ 3 /  + 8*\/ 3                          -4 + 4*\-1 + \/ 3 /  + 8*\/ 3       -4 + 4*\-1 + \/ 3 /  + 8*\/ 3                               -4 + 4*\-1 + \/ 3 /  + 8*\/ 3                                                         -4 + 4*\-1 + \/ 3 /  + 8*\/ 3                               -4 + 4*\-1 + \/ 3 /  + 8*\/ 3    -4 + 4*\-1 + \/ 3 /  + 8*\/ 3
−log⁡(3)−2(−1+3)log⁡(−2+(−1+3)4+4(−1+3)3+6(−1+3)2+43)−4+4(−1+3)2+83−log⁡(−2+(−1+3)4+4(−1+3)3+6(−1+3)2+43)−4+4(−1+3)2+83−(−1+3)2log⁡(−2+(−1+3)4+4(−1+3)3+6(−1+3)2+43)−4+4(−1+3)2+83−atan⁡(19)18+2atan⁡(1−1+(−1+3)2+23)−4+4(−1+3)2+83+4(−1+3)2log⁡(3)−4+4(−1+3)2+83+4log⁡(3)−4+4(−1+3)2+83+8(−1+3)log⁡(3)−4+4(−1+3)2+83+log⁡(82)4- \log{\left(3 \right)} - \frac{2 \left(-1 + \sqrt{3}\right) \log{\left(-2 + \left(-1 + \sqrt{3}\right)^{4} + 4 \left(-1 + \sqrt{3}\right)^{3} + 6 \left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} - \frac{\log{\left(-2 + \left(-1 + \sqrt{3}\right)^{4} + 4 \left(-1 + \sqrt{3}\right)^{3} + 6 \left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} - \frac{\left(-1 + \sqrt{3}\right)^{2} \log{\left(-2 + \left(-1 + \sqrt{3}\right)^{4} + 4 \left(-1 + \sqrt{3}\right)^{3} + 6 \left(-1 + \sqrt{3}\right)^{2} + 4 \sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} - \frac{\operatorname{atan}{\left(\frac{1}{9} \right)}}{18} + \frac{2 \operatorname{atan}{\left(\frac{1}{-1 + \left(-1 + \sqrt{3}\right)^{2} + 2 \sqrt{3}} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{4 \left(-1 + \sqrt{3}\right)^{2} \log{\left(\sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{4 \log{\left(\sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{8 \left(-1 + \sqrt{3}\right) \log{\left(\sqrt{3} \right)}}{-4 + 4 \left(-1 + \sqrt{3}\right)^{2} + 8 \sqrt{3}} + \frac{\log{\left(82 \right)}}{4} 
-log(3) - atan(1/9)/18 + log(82)/4 - log(-2 + (-1 + sqrt(3))^4 + 4*sqrt(3) + 4*(-1 + sqrt(3))^3 + 6*(-1 + sqrt(3))^2)/(-4 + 4*(-1 + sqrt(3))^2 + 8*sqrt(3)) + 2*atan(1/(-1 + (-1 + sqrt(3))^2 + 2*sqrt(3)))/(-4 + 4*(-1 + sqrt(3))^2 + 8*sqrt(3)) + 4*log(sqrt(3))/(-4 + 4*(-1 + sqrt(3))^2 + 8*sqrt(3)) - (-1 + sqrt(3))^2*log(-2 + (-1 + sqrt(3))^4 + 4*sqrt(3) + 4*(-1 + sqrt(3))^3 + 6*(-1 + sqrt(3))^2)/(-4 + 4*(-1 + sqrt(3))^2 + 8*sqrt(3)) - 2*(-1 + sqrt(3))*log(-2 + (-1 + sqrt(3))^4 + 4*sqrt(3) + 4*(-1 + sqrt(3))^3 + 6*(-1 + sqrt(3))^2)/(-4 + 4*(-1 + sqrt(3))^2 + 8*sqrt(3)) + 4*(-1 + sqrt(3))^2*log(sqrt(3))/(-4 + 4*(-1 + sqrt(3))^2 + 8*sqrt(3)) + 8*(-1 + sqrt(3))*log(sqrt(3))/(-4 + 4*(-1 + sqrt(3))^2 + 8*sqrt(3))
Respuesta numérica [src] 
0.0242048632343876
0.0242048632343876

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

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