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Identical expressions
x*log^ three *x
x multiply by logarithm of cubed multiply by x
x multiply by logarithm of to the power of three multiply by x
x*log3*x
x*log³*x
x*log to the power of 3*x
xlog^3x
xlog3x
x*log^3*xdx

Integral of xlog^3x dx

The solution

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  1             
  /             
 |              
 |       3      
 |  x*log (x) dx
 |              
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0

$$\int\limits_{0}^{1} x \log{\left(x \right)}^{3}\, dx$$

Integral(x*log(x)^3, (x, 0, 1))

Detail solution

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

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

$\int u^{3} e^{2 u}\, du$
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(u \right)} = u^{3}$ and let $\operatorname{dv}{\left(u \right)} = e^{2 u}$ .
  
  Then $\operatorname{du}{\left(u \right)} = 3 u^{2}$ .
  
  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 $u{\left(u \right)} = \frac{3 u^{2}}{2}$ and let $\operatorname{dv}{\left(u \right)} = e^{2 u}$ .
  
  Then $\operatorname{du}{\left(u \right)} = 3 u$ .
  
  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.
3. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(u \right)} = \frac{3 u}{2}$ and let $\operatorname{dv}{\left(u \right)} = e^{2 u}$ .
  
  Then $\operatorname{du}{\left(u \right)} = \frac{3}{2}$ .
  
  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.
4. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{3 e^{2 u}}{4}\, du = \frac{3 \int e^{2 u}\, du}{4}$
  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: $\frac{3 e^{2 u}}{8}$
Now substitute $u$ back in:

$\frac{x^{2} \log{\left(x \right)}^{3}}{2} - \frac{3 x^{2} \log{\left(x \right)}^{2}}{4} + \frac{3 x^{2} \log{\left(x \right)}}{4} - \frac{3 x^{2}}{8}$
Now simplify:

$\frac{x^{2} \left(4 \log{\left(x \right)}^{3} - 6 \log{\left(x \right)}^{2} + 6 \log{\left(x \right)} - 3\right)}{8}$
Add the constant of integration:

$\frac{x^{2} \left(4 \log{\left(x \right)}^{3} - 6 \log{\left(x \right)}^{2} + 6 \log{\left(x \right)} - 3\right)}{8}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(4 \log{\left(x \right)}^{3} - 6 \log{\left(x \right)}^{2} + 6 \log{\left(x \right)} - 3\right)}{8}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                 
 |                       2    2    3         2    2         2       
 |      3             3*x    x *log (x)   3*x *log (x)   3*x *log(x)
 | x*log (x) dx = C - ---- + ---------- - ------------ + -----------
 |                     8         2             4              4     
/

$$\int x \log{\left(x \right)}^{3}\, dx = C + \frac{x^{2} \log{\left(x \right)}^{3}}{2} - \frac{3 x^{2} \log{\left(x \right)}^{2}}{4} + \frac{3 x^{2} \log{\left(x \right)}}{4} - \frac{3 x^{2}}{8}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-3/8

$$- \frac{3}{8}$$

-3/8

$$- \frac{3}{8}$$

-3/8

Numerical answer [src]

-0.375

-0.375

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

Other calculators

How to use it?

Identical expressions

Integral of xlog^3x dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Integral of x*log^3*x dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of xlog^3x dx