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Resolución de integrales/ y-y^2/ 1/(k-y-y^2+k*y)

Integral de 1/(k-y-y^2+k*y) dy

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                    
  /                    
 |                     
 |         1           
 |  ---------------- dy
 |           2         
 |  k - y - y  + k*y   
 |                     
/                      
0
∫011ky+(−y2+(k−y)) dy\int\limits_{0}^{1} \frac{1}{k y + \left(- y^{2} + \left(k - y\right)\right)}\, dy 
Integral(1/(k - y - y^2 + k*y), (y, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                        
 |                                                         
 |        1                  log(2 + 2*y)   log(-2*y + 2*k)
 | ---------------- dy = C + ------------ - ---------------
 |          2                   1 + k            1 + k     
 | k - y - y  + k*y                                        
 |                                                         
/
∫1ky+(−y2+(k−y)) dy=C−log⁡(2k−2y)k+1+log⁡(2y+2)k+1\int \frac{1}{k y + \left(- y^{2} + \left(k - y\right)\right)}\, dy = C - \frac{\log{\left(2 k - 2 y \right)}}{k + 1} + \frac{\log{\left(2 y + 2 \right)}}{k + 1}
Simplificar
Respuesta [src] 
   /                                 2   \      /                                 2   \      /                                 2   \      /                                 2   \
   |1   k       1         k         k    |      |3       1       k     k         k    |      |1       1       k     k         k    |      |3   k       1         k         k    |
log|- - - - --------- - ----- - ---------|   log|- + --------- - - + ----- + ---------|   log|- + --------- - - + ----- + ---------|   log|- - - - --------- - ----- - ---------|
   \2   2   2*(1 + k)   1 + k   2*(1 + k)/      \2   2*(1 + k)   2   1 + k   2*(1 + k)/      \2   2*(1 + k)   2   1 + k   2*(1 + k)/      \2   2   2*(1 + k)   1 + k   2*(1 + k)/
------------------------------------------ + ------------------------------------------ - ------------------------------------------ - ------------------------------------------
                  1 + k                                        1 + k                                        1 + k                                        1 + k
log⁡(−k22(k+1)−k2−kk+1+12−12(k+1))k+1−log⁡(−k22(k+1)−k2−kk+1+32−12(k+1))k+1−log⁡(k22(k+1)−k2+kk+1+12+12(k+1))k+1+log⁡(k22(k+1)−k2+kk+1+32+12(k+1))k+1\frac{\log{\left(- \frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} - \frac{k}{k + 1} + \frac{1}{2} - \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1} - \frac{\log{\left(- \frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} - \frac{k}{k + 1} + \frac{3}{2} - \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1} - \frac{\log{\left(\frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} + \frac{k}{k + 1} + \frac{1}{2} + \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1} + \frac{\log{\left(\frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} + \frac{k}{k + 1} + \frac{3}{2} + \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1} 
=
= 
   /                                 2   \      /                                 2   \      /                                 2   \      /                                 2   \
   |1   k       1         k         k    |      |3       1       k     k         k    |      |1       1       k     k         k    |      |3   k       1         k         k    |
log|- - - - --------- - ----- - ---------|   log|- + --------- - - + ----- + ---------|   log|- + --------- - - + ----- + ---------|   log|- - - - --------- - ----- - ---------|
   \2   2   2*(1 + k)   1 + k   2*(1 + k)/      \2   2*(1 + k)   2   1 + k   2*(1 + k)/      \2   2*(1 + k)   2   1 + k   2*(1 + k)/      \2   2   2*(1 + k)   1 + k   2*(1 + k)/
------------------------------------------ + ------------------------------------------ - ------------------------------------------ - ------------------------------------------
                  1 + k                                        1 + k                                        1 + k                                        1 + k
log⁡(−k22(k+1)−k2−kk+1+12−12(k+1))k+1−log⁡(−k22(k+1)−k2−kk+1+32−12(k+1))k+1−log⁡(k22(k+1)−k2+kk+1+12+12(k+1))k+1+log⁡(k22(k+1)−k2+kk+1+32+12(k+1))k+1\frac{\log{\left(- \frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} - \frac{k}{k + 1} + \frac{1}{2} - \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1} - \frac{\log{\left(- \frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} - \frac{k}{k + 1} + \frac{3}{2} - \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1} - \frac{\log{\left(\frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} + \frac{k}{k + 1} + \frac{1}{2} + \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1} + \frac{\log{\left(\frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} + \frac{k}{k + 1} + \frac{3}{2} + \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1} 
log(1/2 - k/2 - 1/(2*(1 + k)) - k/(1 + k) - k^2/(2*(1 + k)))/(1 + k) + log(3/2 + 1/(2*(1 + k)) - k/2 + k/(1 + k) + k^2/(2*(1 + k)))/(1 + k) - log(1/2 + 1/(2*(1 + k)) - k/2 + k/(1 + k) + k^2/(2*(1 + k)))/(1 + k) - log(3/2 - k/2 - 1/(2*(1 + k)) - k/(1 + k) - k^2/(2*(1 + k)))/(1 + k)

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

    © Sr Examen – Калькулятор