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Integral of d{x}:
Integral of |x-1|
Integral of x^2√(1-x^2)
Integral of log(x)*dx/x
Integral of 3^x*e^x
Identical expressions
√x*ln^ two (x)
√x multiply by ln squared (x)
√x multiply by ln to the power of two (x)
√x*ln2(x)
√x*ln2x
√x*ln²(x)
√x*ln to the power of 2(x)
√xln^2(x)
√xln2(x)
√xln2x
√xln^2x
√x*ln^2(x)dx

Integral of √x*ln^2(x) dx

The solution

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  1                 
  /                 
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 |    ___    2      
 |  \/ x *log (x) dx
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0

\int\limits_{0}^{1} \sqrt{x} \log{\left(x \right)}^{2}\, dx

Integral(sqrt(x)*log(x)^2, (x, 0, 1))

Detail solution

Let $u = \log{\left(x \right)}$ .

Then let $du = \frac{dx}{x}$ and substitute $du$ :

$\int u^{2} e^{\frac{3 u}{2}}\, du$
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(u \right)} = u^{2}$ and let $\operatorname{dv}{\left(u \right)} = e^{\frac{3 u}{2}}$ .
  
  Then $\operatorname{du}{\left(u \right)} = 2 u$ .
  
  To find $v{\left(u \right)}$ :
  1. There are multiple ways to do this integral.
    Method #1
    1. Let $u = \frac{3 u}{2}$ .
      
      Then let $du = \frac{3 du}{2}$ and substitute $\frac{2 du}{3}$ :
      
      $\int \frac{4 e^{u}}{9}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{2 e^{u}}{3}\, du = \frac{2 \int e^{u}\, du}{3}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{2 e^{u}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{2 e^{\frac{3 u}{2}}}{3}$
    Method #2
    1. Let $u = e^{\frac{3 u}{2}}$ .
      
      Then let $du = \frac{3 e^{\frac{3 u}{2}} du}{2}$ and substitute $\frac{2 du}{3}$ :
      
      $\int \frac{4}{9}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{2}{3}\, du = \frac{2 \int 1\, du}{3}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{2 u}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{2 e^{\frac{3 u}{2}}}{3}$
  Now evaluate the sub-integral.
2. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(u \right)} = \frac{4 u}{3}$ and let $\operatorname{dv}{\left(u \right)} = e^{\frac{3 u}{2}}$ .
  
  Then $\operatorname{du}{\left(u \right)} = \frac{4}{3}$ .
  
  To find $v{\left(u \right)}$ :
  1. Let $u = \frac{3 u}{2}$ .
    
    Then let $du = \frac{3 du}{2}$ and substitute $\frac{2 du}{3}$ :
    
    $\int \frac{4 e^{u}}{9}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{2 e^{u}}{3}\, du = \frac{2 \int e^{u}\, du}{3}$
      1. The integral of the exponential function is itself.
        
        $\int e^{u}\, du = e^{u}$
      So, the result is: $\frac{2 e^{u}}{3}$
    Now substitute $u$ back in:
    
    $\frac{2 e^{\frac{3 u}{2}}}{3}$
  Now evaluate the sub-integral.
3. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{8 e^{\frac{3 u}{2}}}{9}\, du = \frac{8 \int e^{\frac{3 u}{2}}\, du}{9}$
  1. Let $u = \frac{3 u}{2}$ .
    
    Then let $du = \frac{3 du}{2}$ and substitute $\frac{2 du}{3}$ :
    
    $\int \frac{4 e^{u}}{9}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{2 e^{u}}{3}\, du = \frac{2 \int e^{u}\, du}{3}$
      1. The integral of the exponential function is itself.
        
        $\int e^{u}\, du = e^{u}$
      So, the result is: $\frac{2 e^{u}}{3}$
    Now substitute $u$ back in:
    
    $\frac{2 e^{\frac{3 u}{2}}}{3}$
  So, the result is: $\frac{16 e^{\frac{3 u}{2}}}{27}$
Now substitute $u$ back in:

$\frac{2 x^{\frac{3}{2}} \log{\left(x \right)}^{2}}{3} - \frac{8 x^{\frac{3}{2}} \log{\left(x \right)}}{9} + \frac{16 x^{\frac{3}{2}}}{27}$
Now simplify:

$\frac{2 x^{\frac{3}{2}} \cdot \left(9 \log{\left(x \right)}^{2} - 12 \log{\left(x \right)} + 8\right)}{27}$
Add the constant of integration:

$\frac{2 x^{\frac{3}{2}} \cdot \left(9 \log{\left(x \right)}^{2} - 12 \log{\left(x \right)} + 8\right)}{27}+ \mathrm{constant}$

The answer is:

\frac{2 x^{\frac{3}{2}} \cdot \left(9 \log{\left(x \right)}^{2} - 12 \log{\left(x \right)} + 8\right)}{27}+ \mathrm{constant}

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}}

Simplify

The answer [src]

  1                                                                                                                                                                                    
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 |  /                                                                                                                                                                 /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                                                                                                                                                                                    
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 |  /                                                                                                                                                                 /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                                                
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/                                                                                                                                                                                      
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

Simplify

Numerical answer [src]

0.592592592592593

0.592592592592593

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

Other calculators

How to use it?

Identical expressions

Integral of √x*ln^2(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2