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Resolución de integrales/ (x^2)/ sqrt(x)/ sqrt^3/ sqrt(x)/(3*x-sqrt^3(x^2))

Integral de sqrt(x)/(3*x-sqrt^3(x^2)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                  
  /                  
 |                   
 |        ___        
 |      \/ x         
 |  -------------- dx
 |               3   
 |           ____    
 |          /  2     
 |  3*x - \/  x      
 |                   
/                    
0
∫01x3x−(x2)3 dx\int\limits_{0}^{1} \frac{\sqrt{x}}{3 x - \left(\sqrt{x^{2}}\right)^{3}}\, dx 
Integral(sqrt(x)/(3*x - (sqrt(x^2))^3), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                             / 3/4   ___\                           
 |                                                    4 ___     |3   *\/ x |                           
 |       ___               4 ___    /  ___   4 ___\   \/ 3 *atan|----------|   4 ___    /4 ___     ___\
 |     \/ x                \/ 3 *log\\/ x  - \/ 3 /             \    3     /   \/ 3 *log\\/ 3  + \/ x /
 | -------------- dx = C - ------------------------ + ---------------------- + ------------------------
 |              3                     6                         3                         6            
 |          ____                                                                                       
 |         /  2                                                                                        
 | 3*x - \/  x                                                                                         
 |                                                                                                     
/
∫x3x−(x2)3 dx=C−34log⁡(x−34)6+34log⁡(x+34)6+34atan⁡(334x3)3\int \frac{\sqrt{x}}{3 x - \left(\sqrt{x^{2}}\right)^{3}}\, dx = C - \frac{\sqrt[4]{3} \log{\left(\sqrt{x} - \sqrt[4]{3} \right)}}{6} + \frac{\sqrt[4]{3} \log{\left(\sqrt{x} + \sqrt[4]{3} \right)}}{6} + \frac{\sqrt[4]{3} \operatorname{atan}{\left(\frac{3^{\frac{3}{4}} \sqrt{x}}{3} \right)}}{3}
Simplificar
Respuesta [src] 
                                                                / 3/4\                                                   
                                                      4 ___     |3   |                                                   
  4 ___ /          /     4 ___\\   4 ___    /4 ___\   \/ 3 *atan|----|   4 ___ /          /4 ___\\   4 ___    /    4 ___\
  \/ 3 *\pi*I + log\-1 + \/ 3 //   \/ 3 *log\\/ 3 /             \ 3  /   \/ 3 *\pi*I + log\\/ 3 //   \/ 3 *log\1 + \/ 3 /
- ------------------------------ - ---------------- + ---------------- + ------------------------- + --------------------
                6                         6                  3                       6                        6
−34log⁡(34)6+34log⁡(1+34)6+34atan⁡(3343)3−34(log⁡(−1+34)+iπ)6+34(log⁡(34)+iπ)6- \frac{\sqrt[4]{3} \log{\left(\sqrt[4]{3} \right)}}{6} + \frac{\sqrt[4]{3} \log{\left(1 + \sqrt[4]{3} \right)}}{6} + \frac{\sqrt[4]{3} \operatorname{atan}{\left(\frac{3^{\frac{3}{4}}}{3} \right)}}{3} - \frac{\sqrt[4]{3} \left(\log{\left(-1 + \sqrt[4]{3} \right)} + i \pi\right)}{6} + \frac{\sqrt[4]{3} \left(\log{\left(\sqrt[4]{3} \right)} + i \pi\right)}{6} 
=
= 
                                                                / 3/4\                                                   
                                                      4 ___     |3   |                                                   
  4 ___ /          /     4 ___\\   4 ___    /4 ___\   \/ 3 *atan|----|   4 ___ /          /4 ___\\   4 ___    /    4 ___\
  \/ 3 *\pi*I + log\-1 + \/ 3 //   \/ 3 *log\\/ 3 /             \ 3  /   \/ 3 *\pi*I + log\\/ 3 //   \/ 3 *log\1 + \/ 3 /
- ------------------------------ - ---------------- + ---------------- + ------------------------- + --------------------
                6                         6                  3                       6                        6
−34log⁡(34)6+34log⁡(1+34)6+34atan⁡(3343)3−34(log⁡(−1+34)+iπ)6+34(log⁡(34)+iπ)6- \frac{\sqrt[4]{3} \log{\left(\sqrt[4]{3} \right)}}{6} + \frac{\sqrt[4]{3} \log{\left(1 + \sqrt[4]{3} \right)}}{6} + \frac{\sqrt[4]{3} \operatorname{atan}{\left(\frac{3^{\frac{3}{4}}}{3} \right)}}{3} - \frac{\sqrt[4]{3} \left(\log{\left(-1 + \sqrt[4]{3} \right)} + i \pi\right)}{6} + \frac{\sqrt[4]{3} \left(\log{\left(\sqrt[4]{3} \right)} + i \pi\right)}{6} 
-3^(1/4)*(pi*i + log(-1 + 3^(1/4)))/6 - 3^(1/4)*log(3^(1/4))/6 + 3^(1/4)*atan(3^(3/4)/3)/3 + 3^(1/4)*(pi*i + log(3^(1/4)))/6 + 3^(1/4)*log(1 + 3^(1/4))/6
Respuesta numérica [src] 
0.721907167005821
0.721907167005821

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

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