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Resolución de integrales/ x^2-2/ xln|x^2-2x|

Integral de xln|x^2-2x| dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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∫01xlog⁡(∣x2−2x∣) dx\int\limits_{0}^{1} x \log{\left(\left|{x^{2} - 2 x}\right| \right)}\, dx 
Integral(x*log(|x^2 - 2*x|), (x, 0, 1))
Respuesta (Indefinida) [src] 
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  /                            |     3    d             / 2      \       |     3    d             / 2      \       |     2    d                   / 2      \       |     2    d                   / 2      \         |   d                   / 2      \         |   d                   / 2      \         |   d                         / 2      \         |     2    d             / 2      \                            |     2    d             / 2      \   
 |                             | x*im (x)*--(im(x))*sign\x  - 2*x/       | x*re (x)*--(re(x))*sign\x  - 2*x/       | x*im (x)*--(re(x))*re(x)*sign\x  - 2*x/       | x*re (x)*--(im(x))*im(x)*sign\x  - 2*x/         | x*--(im(x))*im(x)*sign\x  - 2*x/         | x*--(re(x))*re(x)*sign\x  - 2*x/         | x*--(im(x))*im(x)*re(x)*sign\x  - 2*x/         | x*re (x)*--(re(x))*sign\x  - 2*x/       2    /| 2      |\    | x*im (x)*--(re(x))*sign\x  - 2*x/   
 |      /| 2      |\           |          dx                             |          dx                             |          dx                                   |          dx                                     |   dx                                     |   dx                                     |   dx                                           |          dx                            x *log\|x  - 2*x|/    |          dx                         
 | x*log\|x  - 2*x|/ dx = C -  | --------------------------------- dx -  | --------------------------------- dx -  | --------------------------------------- dx -  | --------------------------------------- dx - 2* | -------------------------------- dx - 2* | -------------------------------- dx + 2* | -------------------------------------- dx + 3* | --------------------------------- dx + ------------------ +  | --------------------------------- dx
 |                             |                 | 2      |              |                 | 2      |              |                    | 2      |                 |                    | 2      |                   |                | 2      |                |                | 2      |                |                   | 2      |                   |                 | 2      |                     2             |                 | 2      |          
/                              |        (-2 + x)*|x  - 2*x|              |        (-2 + x)*|x  - 2*x|              |           (-2 + x)*|x  - 2*x|                 |           (-2 + x)*|x  - 2*x|                   |       (-2 + x)*|x  - 2*x|                |       (-2 + x)*|x  - 2*x|                |          (-2 + x)*|x  - 2*x|                   |        (-2 + x)*|x  - 2*x|                                   |        (-2 + x)*|x  - 2*x|          
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∫xlog⁡(∣x2−2x∣) dx=C+x2log⁡(∣x2−2x∣)2−2∫xre⁡(x)sign⁡(x2−2x)ddxre⁡(x)(x−2)∣x2−2x∣ dx+3∫x(re⁡(x))2sign⁡(x2−2x)ddxre⁡(x)(x−2)∣x2−2x∣ dx−∫x(re⁡(x))3sign⁡(x2−2x)ddxre⁡(x)(x−2)∣x2−2x∣ dx−2∫xim⁡(x)sign⁡(x2−2x)ddxim⁡(x)(x−2)∣x2−2x∣ dx+∫x(im⁡(x))2sign⁡(x2−2x)ddxre⁡(x)(x−2)∣x2−2x∣ dx−∫x(im⁡(x))3sign⁡(x2−2x)ddxim⁡(x)(x−2)∣x2−2x∣ dx+2∫xre⁡(x)im⁡(x)sign⁡(x2−2x)ddxim⁡(x)(x−2)∣x2−2x∣ dx−∫xre⁡(x)(im⁡(x))2sign⁡(x2−2x)ddxre⁡(x)(x−2)∣x2−2x∣ dx−∫x(re⁡(x))2im⁡(x)sign⁡(x2−2x)ddxim⁡(x)(x−2)∣x2−2x∣ dx\int x \log{\left(\left|{x^{2} - 2 x}\right| \right)}\, dx = C + \frac{x^{2} \log{\left(\left|{x^{2} - 2 x}\right| \right)}}{2} - 2 \int \frac{x \operatorname{re}{\left(x\right)} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx + 3 \int \frac{x \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx - \int \frac{x \left(\operatorname{re}{\left(x\right)}\right)^{3} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx - 2 \int \frac{x \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx + \int \frac{x \left(\operatorname{im}{\left(x\right)}\right)^{2} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx - \int \frac{x \left(\operatorname{im}{\left(x\right)}\right)^{3} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx + 2 \int \frac{x \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx - \int \frac{x \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{2} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{re}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx - \int \frac{x \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{im}{\left(x\right)} \operatorname{sign}{\left(x^{2} - 2 x \right)} \frac{d}{d x} \operatorname{im}{\left(x\right)}}{\left(x - 2\right) \left|{x^{2} - 2 x}\right|}\, dx
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Respuesta [src] 
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 |  | -1 - x - ------ + x*log\x  - 2*x/ + -------------    for x  - 2*x >= 0   
 |  |          -2 + x                        / 2      \                        
 |  |                                      2*\x  - 2*x/                        
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 |  |           2           /   2      \    x *(2 - 2*x)                       
 |  |-1 - x - ------ + x*log\- x  + 2*x/ + --------------      otherwise       
 |  |         -2 + x                         /   2      \                      
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∫01{x2(2x−2)2(x2−2x)+xlog⁡(x2−2x)−x−1−2x−2for x2−2x≥0x2(2−2x)2(−x2+2x)+xlog⁡(−x2+2x)−x−1−2x−2otherwise dx\int\limits_{0}^{1} \begin{cases} \frac{x^{2} \left(2 x - 2\right)}{2 \left(x^{2} - 2 x\right)} + x \log{\left(x^{2} - 2 x \right)} - x - 1 - \frac{2}{x - 2} & \text{for}\: x^{2} - 2 x \geq 0 \\\frac{x^{2} \left(2 - 2 x\right)}{2 \left(- x^{2} + 2 x\right)} + x \log{\left(- x^{2} + 2 x \right)} - x - 1 - \frac{2}{x - 2} & \text{otherwise} \end{cases}\, dx 
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 |  |            2           / 2      \   x *(-2 + 2*x)         2              
 |  | -1 - x - ------ + x*log\x  - 2*x/ + -------------    for x  - 2*x >= 0   
 |  |          -2 + x                        / 2      \                        
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∫01{x2(2x−2)2(x2−2x)+xlog⁡(x2−2x)−x−1−2x−2for x2−2x≥0x2(2−2x)2(−x2+2x)+xlog⁡(−x2+2x)−x−1−2x−2otherwise dx\int\limits_{0}^{1} \begin{cases} \frac{x^{2} \left(2 x - 2\right)}{2 \left(x^{2} - 2 x\right)} + x \log{\left(x^{2} - 2 x \right)} - x - 1 - \frac{2}{x - 2} & \text{for}\: x^{2} - 2 x \geq 0 \\\frac{x^{2} \left(2 - 2 x\right)}{2 \left(- x^{2} + 2 x\right)} + x \log{\left(- x^{2} + 2 x \right)} - x - 1 - \frac{2}{x - 2} & \text{otherwise} \end{cases}\, dx 
Integral(Piecewise((-1 - x - 2/(-2 + x) + x*log(x^2 - 2*x) + x^2*(-2 + 2*x)/(2*(x^2 - 2*x)), x^2 - 2*x >= 0), (-1 - x - 2/(-2 + x) + x*log(-x^2 + 2*x) + x^2*(2 - 2*x)/(2*(-x^2 + 2*x)), True)), (x, 0, 1))
Respuesta numérica [src] 
-0.113705638880109
-0.113705638880109

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

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