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Integral of d{x}:
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Integral of xln^2x
Integral of 18x^2
Graphing y =:
xln^2x
Identical expressions
xln^2x
xln squared x
xln2x
xln²x
xln to the power of 2x
xln^2xdx
Similar expressions
√x*ln^2(x)
1/(x*ln^2x)

Solving integrals/ xln^2x

Integral of xln^2x dx

The solution

You have entered [src]

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

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

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

Detail solution

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

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

$\int u^{2} e^{2 u}\, du$
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 $\operatorname{dv}{\left(u \right)} = e^{2 u}$ .
  
  Then $\operatorname{du}{\left(u \right)} = 2 u$ .
  
  To find $v{\left(u \right)}$ :
  1. Let $u = 2 u$ .
    
    Then let $du = 2 du$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{e^{u}}{2}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      1. The integral of the exponential function is itself.
        
        $\int e^{u}\, du = e^{u}$
      So, the result is: $\frac{e^{u}}{2}$
    Now substitute $u$ back in:
    
    $\frac{e^{2 u}}{2}$
  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 $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{2 u}$ .
  
  Then $\operatorname{du}{\left(u \right)} = 1$ .
  
  To find $v{\left(u \right)}$ :
  1. Let $u = 2 u$ .
    
    Then let $du = 2 du$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{e^{u}}{2}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      1. The integral of the exponential function is itself.
        
        $\int e^{u}\, du = e^{u}$
      So, the result is: $\frac{e^{u}}{2}$
    Now substitute $u$ back in:
    
    $\frac{e^{2 u}}{2}$
  Now evaluate the sub-integral.
3. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{e^{2 u}}{2}\, du = \frac{\int e^{2 u}\, du}{2}$
  1. Let $u = 2 u$ .
    
    Then let $du = 2 du$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{e^{u}}{2}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      1. The integral of the exponential function is itself.
        
        $\int e^{u}\, du = e^{u}$
      So, the result is: $\frac{e^{u}}{2}$
    Now substitute $u$ back in:
    
    $\frac{e^{2 u}}{2}$
  So, the result is: $\frac{e^{2 u}}{4}$
Now substitute $u$ back in:

$\frac{x^{2} \log{\left(x \right)}^{2}}{2} - \frac{x^{2} \log{\left(x \right)}}{2} + \frac{x^{2}}{4}$
Now simplify:

$\frac{x^{2} \left(2 \log{\left(x \right)}^{2} - 2 \log{\left(x \right)} + 1\right)}{4}$
Add the constant of integration:

$\frac{x^{2} \left(2 \log{\left(x \right)}^{2} - 2 \log{\left(x \right)} + 1\right)}{4}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(2 \log{\left(x \right)}^{2} - 2 \log{\left(x \right)} + 1\right)}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                              
 |                     2    2    2       2       
 |      2             x    x *log (x)   x *log(x)
 | x*log (x) dx = C + -- + ---------- - ---------
 |                    4        2            2    
/

$$\int x \log{\left(x \right)}^{2}\, dx = C + \frac{x^{2} \log{\left(x \right)}^{2}}{2} - \frac{x^{2} \log{\left(x \right)}}{2} + \frac{x^{2}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/4

$$\frac{1}{4}$$

1/4

$$\frac{1}{4}$$

1/4

Numerical answer [src]

0.25

0.25

The graph

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

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Identical expressions

Similar expressions

Integral of xln^2x dx

Limits of integration:

The graph:

Piecewise:

The solution