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Resolución de integrales/ sqrt(x)/ 2+x^2/ sqrt(x)/(2+x^2)

Integral de sqrt(x)/(2+x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
   ___         
 \/ 2          
   /           
  |            
  |     ___    
  |   \/ x     
  |   ------ dx
  |        2   
  |   2 + x    
  |            
 /             
 1
∫12xx2+2 dx\int\limits_{1}^{\sqrt{2}} \frac{\sqrt{x}}{x^{2} + 2}\, dx 
Integral(sqrt(x)/(2 + x^2), (x, 1, sqrt(2)))
Respuesta (Indefinida) [src] 
  /                                                                                                                                                  
 |                                                                                                                                                   
 |   ___           4 ___     /    4 ___   ___\   4 ___     /     4 ___   ___\   4 ___    /      ___    3/4   ___\   4 ___    /      ___    3/4   ___\
 | \/ x            \/ 2 *atan\1 + \/ 2 *\/ x /   \/ 2 *atan\-1 + \/ 2 *\/ x /   \/ 2 *log\x + \/ 2  + 2   *\/ x /   \/ 2 *log\x + \/ 2  - 2   *\/ x /
 | ------ dx = C + --------------------------- + ---------------------------- - --------------------------------- + ---------------------------------
 |      2                       2                             2                                 4                                   4                
 | 2 + x                                                                                                                                             
 |                                                                                                                                                   
/
∫xx2+2 dx=C+24log⁡(−234x+x+2)4−24log⁡(234x+x+2)4+24atan⁡(24x−1)2+24atan⁡(24x+1)2\int \frac{\sqrt{x}}{x^{2} + 2}\, dx = C + \frac{\sqrt[4]{2} \log{\left(- 2^{\frac{3}{4}} \sqrt{x} + x + \sqrt{2} \right)}}{4} - \frac{\sqrt[4]{2} \log{\left(2^{\frac{3}{4}} \sqrt{x} + x + \sqrt{2} \right)}}{4} + \frac{\sqrt[4]{2} \operatorname{atan}{\left(\sqrt[4]{2} \sqrt{x} - 1 \right)}}{2} + \frac{\sqrt[4]{2} \operatorname{atan}{\left(\sqrt[4]{2} \sqrt{x} + 1 \right)}}{2}
Simplificar
Gráfica
1.001.051.101.151.201.251.301.351.400.20.5
Trazar el gráfico f(x)
Respuesta [src] 
4 ___     /    4 ___\   4 ___     /    4 ___\   4 ___    /        ___\   4 ___    /       3/4       ___\      4 ___   4 ___    /         ___\   4 ___    /        ___      3/4\
\/ 2 *atan\1 - \/ 2 /   \/ 2 *atan\1 + \/ 2 /   \/ 2 *log\8 + 8*\/ 2 /   \/ 2 *log\4 - 4*2    + 4*\/ 2 /   pi*\/ 2    \/ 2 *log\-8 + 8*\/ 2 /   \/ 2 *log\4 + 4*\/ 2  + 4*2   /
--------------------- - --------------------- - ---------------------- - ------------------------------- + -------- + ----------------------- + -------------------------------
          2                       2                       4                             4                     4                  4                             4
−24log⁡(8+82)4−24atan⁡(1+24)2−24log⁡(−4⋅234+4+42)4+24atan⁡(1−24)2+24log⁡(−8+82)4+24log⁡(4+42+4⋅234)4+24π4- \frac{\sqrt[4]{2} \log{\left(8 + 8 \sqrt{2} \right)}}{4} - \frac{\sqrt[4]{2} \operatorname{atan}{\left(1 + \sqrt[4]{2} \right)}}{2} - \frac{\sqrt[4]{2} \log{\left(- 4 \cdot 2^{\frac{3}{4}} + 4 + 4 \sqrt{2} \right)}}{4} + \frac{\sqrt[4]{2} \operatorname{atan}{\left(1 - \sqrt[4]{2} \right)}}{2} + \frac{\sqrt[4]{2} \log{\left(-8 + 8 \sqrt{2} \right)}}{4} + \frac{\sqrt[4]{2} \log{\left(4 + 4 \sqrt{2} + 4 \cdot 2^{\frac{3}{4}} \right)}}{4} + \frac{\sqrt[4]{2} \pi}{4} 
=
= 
4 ___     /    4 ___\   4 ___     /    4 ___\   4 ___    /        ___\   4 ___    /       3/4       ___\      4 ___   4 ___    /         ___\   4 ___    /        ___      3/4\
\/ 2 *atan\1 - \/ 2 /   \/ 2 *atan\1 + \/ 2 /   \/ 2 *log\8 + 8*\/ 2 /   \/ 2 *log\4 - 4*2    + 4*\/ 2 /   pi*\/ 2    \/ 2 *log\-8 + 8*\/ 2 /   \/ 2 *log\4 + 4*\/ 2  + 4*2   /
--------------------- - --------------------- - ---------------------- - ------------------------------- + -------- + ----------------------- + -------------------------------
          2                       2                       4                             4                     4                  4                             4
−24log⁡(8+82)4−24atan⁡(1+24)2−24log⁡(−4⋅234+4+42)4+24atan⁡(1−24)2+24log⁡(−8+82)4+24log⁡(4+42+4⋅234)4+24π4- \frac{\sqrt[4]{2} \log{\left(8 + 8 \sqrt{2} \right)}}{4} - \frac{\sqrt[4]{2} \operatorname{atan}{\left(1 + \sqrt[4]{2} \right)}}{2} - \frac{\sqrt[4]{2} \log{\left(- 4 \cdot 2^{\frac{3}{4}} + 4 + 4 \sqrt{2} \right)}}{4} + \frac{\sqrt[4]{2} \operatorname{atan}{\left(1 - \sqrt[4]{2} \right)}}{2} + \frac{\sqrt[4]{2} \log{\left(-8 + 8 \sqrt{2} \right)}}{4} + \frac{\sqrt[4]{2} \log{\left(4 + 4 \sqrt{2} + 4 \cdot 2^{\frac{3}{4}} \right)}}{4} + \frac{\sqrt[4]{2} \pi}{4} 
2^(1/4)*atan(1 - 2^(1/4))/2 - 2^(1/4)*atan(1 + 2^(1/4))/2 - 2^(1/4)*log(8 + 8*sqrt(2))/4 - 2^(1/4)*log(4 - 4*2^(3/4) + 4*sqrt(2))/4 + pi*2^(1/4)/4 + 2^(1/4)*log(-8 + 8*sqrt(2))/4 + 2^(1/4)*log(4 + 4*sqrt(2) + 4*2^(3/4))/4
Respuesta numérica [src] 
0.131299925493769
0.131299925493769

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

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