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Solving integrals/ lnx/sqrtx

Integral of lnx/sqrtx dx

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  1          
  /          
 |           
 |  log(x)   
 |  ------ dx
 |    ___    
 |  \/ x     
 |           
/            
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∫01log⁡(x)x dx\int\limits_{0}^{1} \frac{\log{\left(x \right)}}{\sqrt{x}}\, dx 
Integral(log(x)/(sqrt(x)), (x, 0, 1))
The answer (Indefinite) [src] 
  /                                        
 |                                         
 | log(x)              ___       ___       
 | ------ dx = C - 4*\/ x  + 2*\/ x *log(x)
 |   ___                                   
 | \/ x                                    
 |                                         
/
∫log⁡(x)x dx=C+2xlog⁡(x)−4x\int \frac{\log{\left(x \right)}}{\sqrt{x}}\, dx = C + 2 \sqrt{x} \log{\left(x \right)} - 4 \sqrt{x}
Simplify
The answer [src] 
  1                                                                                                                                              
  /                                                                                                                                              
 |                                                                                                                                               
 |  /                                                     2*log(x)                                                              /1           \   
 |  |                                                     --------                                                       for And|- < 1, x < 1|   
 |  |                                                        ___                                                                \x           /   
 |  |                                                      \/ x                                                                                  
 |  |                                                                                                                                            
 |  |                                                      log(x)                                                              /1           \    
 |  |                                                      ------                                                        for Or|- < 1, x < 1|    
 |  |                                                        ___                                                               \x           /    
 |  <                                                      \/ x                                                                                dx
 |  |                                                                                                                                            
 |  | __0, 3 /3/2, 3/2, 1              |  \                                                                                                      
 |  |/__     |                         | x|                                                                                                      
 |  |\_|3, 3 \             1/2, 1/2, 0 |  /    __0, 3 /1/2, 3/2, 1              |  \    __2, 1 /   0      3/2, 3/2 |  \                          
 |  |-------------------------------------- + /__     |                         | x|   /__     |                   | x|                          
 |  |                  2                      \_|3, 3 \             1/2, 1/2, 0 |  /   \_|3, 3 \1/2, 1/2     0     |  /                          
 |  |------------------------------------------------------------------------------- - --------------------------------        otherwise         
 |  \                                       x                                                         x                                          
 |                                                                                                                                               
/                                                                                                                                                
0
∫01{2log⁡(x)xfor 1x<1∧x<1log⁡(x)xfor 1x<1∨x<1G3,30,3(12,32,112,12,0|x)+G3,30,3(32,32,112,12,0|x)2x−G3,32,1(032,3212,120|x)xotherwise dx\int\limits_{0}^{1} \begin{cases} \frac{2 \log{\left(x \right)}}{\sqrt{x}} & \text{for}\: \frac{1}{x} < 1 \wedge x < 1 \\\frac{\log{\left(x \right)}}{\sqrt{x}} & \text{for}\: \frac{1}{x} < 1 \vee x < 1 \\\frac{{G_{3, 3}^{0, 3}\left(\begin{matrix} \frac{1}{2}, \frac{3}{2}, 1 & \\ & \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle| {x} \right)} + \frac{{G_{3, 3}^{0, 3}\left(\begin{matrix} \frac{3}{2}, \frac{3}{2}, 1 & \\ & \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle| {x} \right)}}{2}}{x} - \frac{{G_{3, 3}^{2, 1}\left(\begin{matrix} 0 & \frac{3}{2}, \frac{3}{2} \\\frac{1}{2}, \frac{1}{2} & 0 \end{matrix} \middle| {x} \right)}}{x} & \text{otherwise} \end{cases}\, dx 
=
= 
  1                                                                                                                                              
  /                                                                                                                                              
 |                                                                                                                                               
 |  /                                                     2*log(x)                                                              /1           \   
 |  |                                                     --------                                                       for And|- < 1, x < 1|   
 |  |                                                        ___                                                                \x           /   
 |  |                                                      \/ x                                                                                  
 |  |                                                                                                                                            
 |  |                                                      log(x)                                                              /1           \    
 |  |                                                      ------                                                        for Or|- < 1, x < 1|    
 |  |                                                        ___                                                               \x           /    
 |  <                                                      \/ x                                                                                dx
 |  |                                                                                                                                            
 |  | __0, 3 /3/2, 3/2, 1              |  \                                                                                                      
 |  |/__     |                         | x|                                                                                                      
 |  |\_|3, 3 \             1/2, 1/2, 0 |  /    __0, 3 /1/2, 3/2, 1              |  \    __2, 1 /   0      3/2, 3/2 |  \                          
 |  |-------------------------------------- + /__     |                         | x|   /__     |                   | x|                          
 |  |                  2                      \_|3, 3 \             1/2, 1/2, 0 |  /   \_|3, 3 \1/2, 1/2     0     |  /                          
 |  |------------------------------------------------------------------------------- - --------------------------------        otherwise         
 |  \                                       x                                                         x                                          
 |                                                                                                                                               
/                                                                                                                                                
0
∫01{2log⁡(x)xfor 1x<1∧x<1log⁡(x)xfor 1x<1∨x<1G3,30,3(12,32,112,12,0|x)+G3,30,3(32,32,112,12,0|x)2x−G3,32,1(032,3212,120|x)xotherwise dx\int\limits_{0}^{1} \begin{cases} \frac{2 \log{\left(x \right)}}{\sqrt{x}} & \text{for}\: \frac{1}{x} < 1 \wedge x < 1 \\\frac{\log{\left(x \right)}}{\sqrt{x}} & \text{for}\: \frac{1}{x} < 1 \vee x < 1 \\\frac{{G_{3, 3}^{0, 3}\left(\begin{matrix} \frac{1}{2}, \frac{3}{2}, 1 & \\ & \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle| {x} \right)} + \frac{{G_{3, 3}^{0, 3}\left(\begin{matrix} \frac{3}{2}, \frac{3}{2}, 1 & \\ & \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle| {x} \right)}}{2}}{x} - \frac{{G_{3, 3}^{2, 1}\left(\begin{matrix} 0 & \frac{3}{2}, \frac{3}{2} \\\frac{1}{2}, \frac{1}{2} & 0 \end{matrix} \middle| {x} \right)}}{x} & \text{otherwise} \end{cases}\, dx
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Numerical answer [src] 
-3.99999997553465
-3.99999997553465

    Use the examples entering the upper and lower limits of integration.

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