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Integral of d{x}:
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Integral of dx/tan5x
Integral of (3-sin(x))/(2cos(x)+3tan(x))
Integral of 1/(x^4+4x^2+4)
Graphing y =:
(x^2)*(ln(x))^2
Derivative of:
(x^2)*(ln(x))^2
Identical expressions
(x^ two)*(ln(x))^ two
(x squared ) multiply by (ln(x)) squared
(x to the power of two) multiply by (ln(x)) to the power of two
(x2)*(ln(x))2
x2*lnx2
(x²)*(ln(x))²
(x to the power of 2)*(ln(x)) to the power of 2
(x^2)(ln(x))^2
(x2)(ln(x))2
x2lnx2
x^2lnx^2
(x^2)*(ln(x))^2dx

Integral of (x^2)*(ln(x))^2 dx

The solution

You have entered [src]

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

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

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

Detail solution

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

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

$\int u^{2} e^{3 u}\, du$
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 $\operatorname{dv}{\left(u \right)} = e^{3 u}$ .
  
  Then $\operatorname{du}{\left(u \right)} = 2 u$ .
  
  To find $v{\left(u \right)}$ :
  1. Let $u = 3 u$ .
    
    Then let $du = 3 du$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{e^{u}}{3}\, du$
    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: $\frac{e^{u}}{3}$
    Now substitute $u$ back in:
    
    $\frac{e^{3 u}}{3}$
  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 $u{\left(u \right)} = \frac{2 u}{3}$ and let $\operatorname{dv}{\left(u \right)} = e^{3 u}$ .
  
  Then $\operatorname{du}{\left(u \right)} = \frac{2}{3}$ .
  
  To find $v{\left(u \right)}$ :
  1. Let $u = 3 u$ .
    
    Then let $du = 3 du$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{e^{u}}{3}\, du$
    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: $\frac{e^{u}}{3}$
    Now substitute $u$ back in:
    
    $\frac{e^{3 u}}{3}$
  Now evaluate the sub-integral.
3. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 e^{3 u}}{9}\, du = \frac{2 \int e^{3 u}\, du}{9}$
  1. Let $u = 3 u$ .
    
    Then let $du = 3 du$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{e^{u}}{3}\, du$
    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: $\frac{e^{u}}{3}$
    Now substitute $u$ back in:
    
    $\frac{e^{3 u}}{3}$
  So, the result is: $\frac{2 e^{3 u}}{27}$
Now substitute $u$ back in:

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

$\frac{x^{3} \left(9 \log{\left(x \right)}^{2} - 6 \log{\left(x \right)} + 2\right)}{27}$
Add the constant of integration:

$\frac{x^{3} \left(9 \log{\left(x \right)}^{2} - 6 \log{\left(x \right)} + 2\right)}{27}+ \mathrm{constant}$

The answer is:

$\frac{x^{3} \left(9 \log{\left(x \right)}^{2} - 6 \log{\left(x \right)} + 2\right)}{27}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                   
 |                        3      3           3    2   
 |  2    2             2*x    2*x *log(x)   x *log (x)
 | x *log (x) dx = C + ---- - ----------- + ----------
 |                      27         9            3     
/

$$\int x^{2} \log{\left(x \right)}^{2}\, dx = C + \frac{x^{3} \log{\left(x \right)}^{2}}{3} - \frac{2 x^{3} \log{\left(x \right)}}{9} + \frac{2 x^{3}}{27}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   3
5*e 
----
 27

$$\frac{5 e^{3}}{27}$$

   3
5*e 
----
 27

$$\frac{5 e^{3}}{27}$$

5*exp(3)/27

Numerical answer [src]

3.71954387466438

3.71954387466438

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

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How to use it?

Identical expressions

Integral of (x^2)*(ln(x))^2 dx

Limits of integration:

The graph:

Piecewise:

The solution