Sr Examen

Otras calculadoras

Resolución de integrales/ arctg/ ctg2x/ (x-arctg2x)/(1-4x^2)

Integral de (x-arctg2x)/(1-4x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
 1/2                
  /                 
 |                  
 |  x - atan(2*x)   
 |  ------------- dx
 |            2     
 |     1 - 4*x      
 |                  
/                   
0
∫012x−atan⁡(2x)1−4x2 dx\int\limits_{0}^{\frac{1}{2}} \frac{x - \operatorname{atan}{\left(2 x \right)}}{1 - 4 x^{2}}\, dx 
Integral((x - atan(2*x))/(1 - 4*x^2), (x, 0, 1/2))
Respuesta (Indefinida) [src] 
                            //                                           /                                                                    \                                                           
                            ||                                          |                                                                     |                                                           
                            ||                                          | acoth(2*x)                                                          |                                                           
                            ||                                        - | ---------- dx                                                       |                                                           
                            ||                                          |         2                                                           |                                                           
                            ||                                          |  1 + 4*x                                                            |                                                           
                            ||                                          |                                                                     |                                                           
                            ||                                         /                                                                      |                                                           
                            ||                                        ------------------                                          for x < -1/2|                                                           
                            ||                                                2                                                               |                                                           
                            ||                                                                                                                |                                                           
                            ||                     -1/2                 -1/2                                                                  |                                                           
                            ||                       /                    /                                                                   |                                                           
                            ||                      |                    |                    /                                               |                                                           
                            ||                      |  atanh(2*x)        |  acoth(2*x)       |                                                |                                                           
  /                         ||                      |  ---------- dx     |  ---------- dx    | atanh(2*x)                                     |                    //-acoth(2*x)        2      \          
 |                          ||                      |          2         |          2        | ---------- dx                                  |      /        2\   ||------------  for x  > 1/4|          
 | x - atan(2*x)            ||                      |   1 + 4*x          |   1 + 4*x         |         2                                      |   log\-1 + 4*x /   ||     2                    |          
 | ------------- dx = C - 2*|<                      |                    |                   |  1 + 4*x                                       | - -------------- + |<                          |*atan(2*x)
 |           2              ||                     /                    /                    |                                                |         8          ||-atanh(2*x)        2      |          
 |    1 - 4*x               ||                                                              /                                                 |                    ||------------  for x  < 1/4|          
 |                          ||                    ------------------ - ------------------ - ----------------                      for x < 1/2 |                    \\     2                    /          
/                           ||                            2                    2                   2                                          |                                                           
                            ||                                                                                                                |                                                           
                            || 1/2                 -1/2                                    -1/2                 1/2                           |                                                           
                            ||  /                    /                                       /                   /                            |                                                           
                            || |                    |                    /                  |                   |                             |                                                           
                            || |  acoth(2*x)        |  atanh(2*x)       |                   |  acoth(2*x)       |  atanh(2*x)                 |                                                           
                            || |  ---------- dx     |  ---------- dx    | acoth(2*x)        |  ---------- dx    |  ---------- dx              |                                                           
                            || |          2         |          2        | ---------- dx     |          2        |          2                  |                                                           
                            || |   1 + 4*x          |   1 + 4*x         |         2         |   1 + 4*x         |   1 + 4*x                   |                                                           
                            || |                    |                   |  1 + 4*x          |                   |                             |                                                           
                            ||/                    /                    |                  /                   /                              |                                                           
                            ||                                         /                                                                      |                                                           
                            ||----------------- + ------------------ - ---------------- - ------------------ - -----------------   otherwise  |                                                           
                            ||        2                   2                   2                   2                    2                      |                                                           
                            \\                                                                                                                /
∫x−atan⁡(2x)1−4x2 dx=C+({−acoth⁡(2x)2for x2>14−atanh⁡(2x)2for x2<14)atan⁡(2x)−2({−∫acoth⁡(2x)4x2+1 dx2for x<−12−∫−12acoth⁡(2x)4x2+1 dx2−∫atanh⁡(2x)4x2+1 dx2+∫−12atanh⁡(2x)4x2+1 dx2for x<12−∫acoth⁡(2x)4x2+1 dx2−∫−12acoth⁡(2x)4x2+1 dx2+∫12acoth⁡(2x)4x2+1 dx2+∫−12atanh⁡(2x)4x2+1 dx2−∫12atanh⁡(2x)4x2+1 dx2otherwise)−log⁡(4x2−1)8\int \frac{x - \operatorname{atan}{\left(2 x \right)}}{1 - 4 x^{2}}\, dx = C + \left(\begin{cases} - \frac{\operatorname{acoth}{\left(2 x \right)}}{2} & \text{for}\: x^{2} > \frac{1}{4} \\- \frac{\operatorname{atanh}{\left(2 x \right)}}{2} & \text{for}\: x^{2} < \frac{1}{4} \end{cases}\right) \operatorname{atan}{\left(2 x \right)} - 2 \left(\begin{cases} - \frac{\int \frac{\operatorname{acoth}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} & \text{for}\: x < - \frac{1}{2} \\- \frac{\int\limits^{- \frac{1}{2}} \frac{\operatorname{acoth}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} - \frac{\int \frac{\operatorname{atanh}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} + \frac{\int\limits^{- \frac{1}{2}} \frac{\operatorname{atanh}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} & \text{for}\: x < \frac{1}{2} \\- \frac{\int \frac{\operatorname{acoth}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} - \frac{\int\limits^{- \frac{1}{2}} \frac{\operatorname{acoth}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} + \frac{\int\limits^{\frac{1}{2}} \frac{\operatorname{acoth}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} + \frac{\int\limits^{- \frac{1}{2}} \frac{\operatorname{atanh}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} - \frac{\int\limits^{\frac{1}{2}} \frac{\operatorname{atanh}{\left(2 x \right)}}{4 x^{2} + 1}\, dx}{2} & \text{otherwise} \end{cases}\right) - \frac{\log{\left(4 x^{2} - 1 \right)}}{8}
Simplificar
Respuesta numérica [src] 
-3.13962065197508
-3.13962065197508

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

    © Sr Examen – Калькулятор