Mister Exam

Other calculators

Integral of √x*ln^2(x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                 
  /                 
 |                  
 |    ___    2      
 |  \/ x *log (x) dx
 |                  
/                   
0                   
$$\int\limits_{0}^{1} \sqrt{x} \log{\left(x \right)}^{2}\, dx$$
Integral(sqrt(x)*log(x)^2, (x, 0, 1))
Detail solution
  1. Let .

    Then let and substitute :

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. There are multiple ways to do this integral.

        Method #1

        1. Let .

          Then let and substitute :

          1. The integral of a constant times a function is the constant times the integral of the function:

            1. The integral of the exponential function is itself.

            So, the result is:

          Now substitute back in:

        Method #2

        1. Let .

          Then let and substitute :

          1. The integral of a constant times a function is the constant times the integral of the function:

            1. The integral of a constant is the constant times the variable of integration:

            So, the result is:

          Now substitute back in:

      Now evaluate the sub-integral.

    2. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. Let .

        Then let and substitute :

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. The integral of the exponential function is itself.

          So, the result is:

        Now substitute back in:

      Now evaluate the sub-integral.

    3. The integral of a constant times a function is the constant times the integral of the function:

      1. Let .

        Then let and substitute :

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. The integral of the exponential function is itself.

          So, the result is:

        Now substitute back in:

      So, the result is:

    Now substitute back in:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                               
 |                            3/2      3/2             3/2    2   
 |   ___    2             16*x      8*x   *log(x)   2*x   *log (x)
 | \/ x *log (x) dx = C + ------- - ------------- + --------------
 |                           27           9               3       
/                                                                 
$${{8\,x^{{{3}\over{2}}}\,\left({{9\,\left(\log x\right)^2}\over{4}}- 3\,\log x+2\right)}\over{27}}$$
The answer [src]
  1                                                                                                                                                                                    
  /                                                                                                                                                                                    
 |                                                                                                                                                                                     
 |  /                                                                                                                                                                 /1           \   
 |  |                                                                           0                                                                              for And|- < 1, x < 1|   
 |  |                                                                                                                                                                 \x           /   
 |  |                                                                                                                                                                                  
 |  |                                                                       ___    2                                                                                 /1           \    
 |  |                                                                     \/ x *log (x)                                                                        for Or|- < 1, x < 1|    
 |  |                                                                                                                                                                \x           /    
 |  <                                                                                                                                                                                dx
 |  |  /   __0, 4 /5/2, 5/2, 5/2, 1                   |  \                                                   \                                                                         
 |  |  |3*/__     |                                   | x|                                                   |                                                                         
 |  |  |  \_|4, 4 \                  3/2, 3/2, 3/2, 0 |  /    __0, 4 /3/2, 5/2, 5/2, 1                   |  \|      __3, 1 /      0        5/2, 5/2, 5/2 |  \                          
 |  |2*|-------------------------------------------------- + /__     |                                   | x||   2*/__     |                             | x|                          
 |  |  \                        2                            \_|4, 4 \                  3/2, 3/2, 3/2, 0 |  //     \_|4, 4 \3/2, 3/2, 3/2        0       |  /                          
 |  |--------------------------------------------------------------------------------------------------------- + --------------------------------------------        otherwise         
 |  \                                                    x                                                                            x                                                
 |                                                                                                                                                                                     
/                                                                                                                                                                                      
0                                                                                                                                                                                      
$${{16}\over{27}}$$
=
=
  1                                                                                                                                                                                    
  /                                                                                                                                                                                    
 |                                                                                                                                                                                     
 |  /                                                                                                                                                                 /1           \   
 |  |                                                                           0                                                                              for And|- < 1, x < 1|   
 |  |                                                                                                                                                                 \x           /   
 |  |                                                                                                                                                                                  
 |  |                                                                       ___    2                                                                                 /1           \    
 |  |                                                                     \/ x *log (x)                                                                        for Or|- < 1, x < 1|    
 |  |                                                                                                                                                                \x           /    
 |  <                                                                                                                                                                                dx
 |  |  /   __0, 4 /5/2, 5/2, 5/2, 1                   |  \                                                   \                                                                         
 |  |  |3*/__     |                                   | x|                                                   |                                                                         
 |  |  |  \_|4, 4 \                  3/2, 3/2, 3/2, 0 |  /    __0, 4 /3/2, 5/2, 5/2, 1                   |  \|      __3, 1 /      0        5/2, 5/2, 5/2 |  \                          
 |  |2*|-------------------------------------------------- + /__     |                                   | x||   2*/__     |                             | x|                          
 |  |  \                        2                            \_|4, 4 \                  3/2, 3/2, 3/2, 0 |  //     \_|4, 4 \3/2, 3/2, 3/2        0       |  /                          
 |  |--------------------------------------------------------------------------------------------------------- + --------------------------------------------        otherwise         
 |  \                                                    x                                                                            x                                                
 |                                                                                                                                                                                     
/                                                                                                                                                                                      
0                                                                                                                                                                                      
$$\int\limits_{0}^{1} \begin{cases} 0 & \text{for}\: \frac{1}{x} < 1 \wedge x < 1 \\\sqrt{x} \log{\left(x \right)}^{2} & \text{for}\: \frac{1}{x} < 1 \vee x < 1 \\\frac{2 \left({G_{4, 4}^{0, 4}\left(\begin{matrix} \frac{3}{2}, \frac{5}{2}, \frac{5}{2}, 1 & \\ & \frac{3}{2}, \frac{3}{2}, \frac{3}{2}, 0 \end{matrix} \middle| {x} \right)} + \frac{3 {G_{4, 4}^{0, 4}\left(\begin{matrix} \frac{5}{2}, \frac{5}{2}, \frac{5}{2}, 1 & \\ & \frac{3}{2}, \frac{3}{2}, \frac{3}{2}, 0 \end{matrix} \middle| {x} \right)}}{2}\right)}{x} + \frac{2 {G_{4, 4}^{3, 1}\left(\begin{matrix} 0 & \frac{5}{2}, \frac{5}{2}, \frac{5}{2} \\\frac{3}{2}, \frac{3}{2}, \frac{3}{2} & 0 \end{matrix} \middle| {x} \right)}}{x} & \text{otherwise} \end{cases}\, dx$$
Numerical answer [src]
0.592592592592593
0.592592592592593

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