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Resolución de integrales/ x+x^2/ 1/(a*b-a*x-b*x+x^2)

Integral de 1/(a*b-a*x-b*x+x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  x                        
  /                        
 |                         
 |           1             
 |  -------------------- dx
 |                     2   
 |  a*b - a*x - b*x + x    
 |                         
/                          
0
∫0x1x2+(−bx+(ab−ax)) dx\int\limits_{0}^{x} \frac{1}{x^{2} + \left(- b x + \left(a b - a x\right)\right)}\, dx 
Integral(1/(a*b - a*x - b*x + x^2), (x, 0, x))
Respuesta (Indefinida) [src] 
  /                                                               
 |                                                                
 |          1                    log(-2*x + 2*a)   log(-2*b + 2*x)
 | -------------------- dx = C + --------------- - ---------------
 |                    2               a - b             a - b     
 | a*b - a*x - b*x + x                                            
 |                                                                
/
∫1x2+(−bx+(ab−ax)) dx=C+log⁡(2a−2x)a−b−log⁡(−2b+2x)a−b\int \frac{1}{x^{2} + \left(- b x + \left(a b - a x\right)\right)}\, dx = C + \frac{\log{\left(2 a - 2 x \right)}}{a - b} - \frac{\log{\left(- 2 b + 2 x \right)}}{a - b}
Simplificar
Respuesta [src] 
   /               2           2           \      /                 2           2           \      /               2           2           \      /                 2           2           \
   |  a   b       a           b        a*b |      |    a   b       a           b        a*b |      |  a   b       a           b        a*b |      |    a   b       a           b        a*b |
log|- - - - + --------- + --------- - -----|   log|x - - - - - --------- - --------- + -----|   log|- - - - - --------- - --------- + -----|   log|x - - - - + --------- + --------- - -----|
   \  2   2   2*(a - b)   2*(a - b)   a - b/      \    2   2   2*(a - b)   2*(a - b)   a - b/      \  2   2   2*(a - b)   2*(a - b)   a - b/      \    2   2   2*(a - b)   2*(a - b)   a - b/
-------------------------------------------- + ---------------------------------------------- - -------------------------------------------- - ----------------------------------------------
                   a - b                                           a - b                                           a - b                                           a - b
−log⁡(−a22(a−b)+aba−b−a2−b22(a−b)−b2)a−b+log⁡(a22(a−b)−aba−b−a2+b22(a−b)−b2)a−b+log⁡(−a22(a−b)+aba−b−a2−b22(a−b)−b2+x)a−b−log⁡(a22(a−b)−aba−b−a2+b22(a−b)−b2+x)a−b- \frac{\log{\left(- \frac{a^{2}}{2 \left(a - b\right)} + \frac{a b}{a - b} - \frac{a}{2} - \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} \right)}}{a - b} + \frac{\log{\left(\frac{a^{2}}{2 \left(a - b\right)} - \frac{a b}{a - b} - \frac{a}{2} + \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} \right)}}{a - b} + \frac{\log{\left(- \frac{a^{2}}{2 \left(a - b\right)} + \frac{a b}{a - b} - \frac{a}{2} - \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} + x \right)}}{a - b} - \frac{\log{\left(\frac{a^{2}}{2 \left(a - b\right)} - \frac{a b}{a - b} - \frac{a}{2} + \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} + x \right)}}{a - b} 
=
= 
   /               2           2           \      /                 2           2           \      /               2           2           \      /                 2           2           \
   |  a   b       a           b        a*b |      |    a   b       a           b        a*b |      |  a   b       a           b        a*b |      |    a   b       a           b        a*b |
log|- - - - + --------- + --------- - -----|   log|x - - - - - --------- - --------- + -----|   log|- - - - - --------- - --------- + -----|   log|x - - - - + --------- + --------- - -----|
   \  2   2   2*(a - b)   2*(a - b)   a - b/      \    2   2   2*(a - b)   2*(a - b)   a - b/      \  2   2   2*(a - b)   2*(a - b)   a - b/      \    2   2   2*(a - b)   2*(a - b)   a - b/
-------------------------------------------- + ---------------------------------------------- - -------------------------------------------- - ----------------------------------------------
                   a - b                                           a - b                                           a - b                                           a - b
−log⁡(−a22(a−b)+aba−b−a2−b22(a−b)−b2)a−b+log⁡(a22(a−b)−aba−b−a2+b22(a−b)−b2)a−b+log⁡(−a22(a−b)+aba−b−a2−b22(a−b)−b2+x)a−b−log⁡(a22(a−b)−aba−b−a2+b22(a−b)−b2+x)a−b- \frac{\log{\left(- \frac{a^{2}}{2 \left(a - b\right)} + \frac{a b}{a - b} - \frac{a}{2} - \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} \right)}}{a - b} + \frac{\log{\left(\frac{a^{2}}{2 \left(a - b\right)} - \frac{a b}{a - b} - \frac{a}{2} + \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} \right)}}{a - b} + \frac{\log{\left(- \frac{a^{2}}{2 \left(a - b\right)} + \frac{a b}{a - b} - \frac{a}{2} - \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} + x \right)}}{a - b} - \frac{\log{\left(\frac{a^{2}}{2 \left(a - b\right)} - \frac{a b}{a - b} - \frac{a}{2} + \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} + x \right)}}{a - b} 
log(-a/2 - b/2 + a^2/(2*(a - b)) + b^2/(2*(a - b)) - a*b/(a - b))/(a - b) + log(x - a/2 - b/2 - a^2/(2*(a - b)) - b^2/(2*(a - b)) + a*b/(a - b))/(a - b) - log(-a/2 - b/2 - a^2/(2*(a - b)) - b^2/(2*(a - b)) + a*b/(a - b))/(a - b) - log(x - a/2 - b/2 + a^2/(2*(a - b)) + b^2/(2*(a - b)) - a*b/(a - b))/(a - b)

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

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