Integral of x²ln4x dx

The solution

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  1               
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 |   2            
 |  x *log(4*x) dx
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0

$$\int\limits_{0}^{1} x^{2} \log{\left(4 x \right)}\, dx$$

Integral(x^2*log(4*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $x^{2} \log{\left(4 x \right)} = x^{2} \log{\left(x \right)} + 2 x^{2} \log{\left(2 \right)}$
2. Integrate term-by-term:
  1. Let $u = \log{\left(x \right)}$ .
    
    Then let $du = \frac{dx}{x}$ and substitute $du$ :
    
    $\int u e^{3 u}\, du$
    1. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{3 u}$ .
      
      Then $\operatorname{du}{\left(u \right)} = 1$ .
      
      To find $v{\left(u \right)}$ :
      
      Let $u = 3 u$ .
      
      Then let $du = 3 du$ and substitute $\frac{du}{3}$ :
      
      $\int \frac{e^{u}}{3}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      
      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. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{3 u}}{3}\, du = \frac{\int e^{3 u}\, du}{3}$
      
      Let $u = 3 u$ .
      
      Then let $du = 3 du$ and substitute $\frac{du}{3}$ :
      
      $\int \frac{e^{u}}{3}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      
      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: $\frac{e^{3 u}}{9}$
    Now substitute $u$ back in:
    
    $\frac{x^{3} \log{\left(x \right)}}{3} - \frac{x^{3}}{9}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 x^{2} \log{\left(2 \right)}\, dx = 2 \log{\left(2 \right)} \int x^{2}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    So, the result is: $\frac{2 x^{3} \log{\left(2 \right)}}{3}$
  The result is: $\frac{x^{3} \log{\left(x \right)}}{3} - \frac{x^{3}}{9} + \frac{2 x^{3} \log{\left(2 \right)}}{3}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \log{\left(4 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = x^{2}$ .
  
  Then $\operatorname{du}{\left(x \right)} = \frac{1}{x}$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x^{2}}{3}\, dx = \frac{\int x^{2}\, dx}{3}$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  So, the result is: $\frac{x^{3}}{9}$
Method #3
1. Rewrite the integrand:
  
  $x^{2} \log{\left(4 x \right)} = x^{2} \log{\left(x \right)} + 2 x^{2} \log{\left(2 \right)}$
2. Integrate term-by-term:
  1. Let $u = \log{\left(x \right)}$ .
    
    Then let $du = \frac{dx}{x}$ and substitute $du$ :
    
    $\int u e^{3 u}\, du$
    1. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{3 u}$ .
      
      Then $\operatorname{du}{\left(u \right)} = 1$ .
      
      To find $v{\left(u \right)}$ :
      
      Let $u = 3 u$ .
      
      Then let $du = 3 du$ and substitute $\frac{du}{3}$ :
      
      $\int \frac{e^{u}}{3}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      
      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. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{3 u}}{3}\, du = \frac{\int e^{3 u}\, du}{3}$
      
      Let $u = 3 u$ .
      
      Then let $du = 3 du$ and substitute $\frac{du}{3}$ :
      
      $\int \frac{e^{u}}{3}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      
      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: $\frac{e^{3 u}}{9}$
    Now substitute $u$ back in:
    
    $\frac{x^{3} \log{\left(x \right)}}{3} - \frac{x^{3}}{9}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 x^{2} \log{\left(2 \right)}\, dx = 2 \log{\left(2 \right)} \int x^{2}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    So, the result is: $\frac{2 x^{3} \log{\left(2 \right)}}{3}$
  The result is: $\frac{x^{3} \log{\left(x \right)}}{3} - \frac{x^{3}}{9} + \frac{2 x^{3} \log{\left(2 \right)}}{3}$
Now simplify:

$\frac{x^{3} \left(3 \log{\left(x \right)} - 1 + \log{\left(64 \right)}\right)}{9}$
Add the constant of integration:

$\frac{x^{3} \left(3 \log{\left(x \right)} - 1 + \log{\left(64 \right)}\right)}{9}+ \mathrm{constant}$

The answer is:

$\frac{x^{3} \left(3 \log{\left(x \right)} - 1 + \log{\left(64 \right)}\right)}{9}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                 
 |                       3    3             3       
 |  2                   x    x *log(x)   2*x *log(2)
 | x *log(4*x) dx = C - -- + --------- + -----------
 |                      9        3            3     
/

$$\int x^{2} \log{\left(4 x \right)}\, dx = C + \frac{x^{3} \log{\left(x \right)}}{3} - \frac{x^{3}}{9} + \frac{2 x^{3} \log{\left(2 \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1   log(4)
- - + ------
  9     3

$$- \frac{1}{9} + \frac{\log{\left(4 \right)}}{3}$$

  1   log(4)
- - + ------
  9     3

$$- \frac{1}{9} + \frac{\log{\left(4 \right)}}{3}$$

-1/9 + log(4)/3

Numerical answer [src]

0.350987009262186

0.350987009262186

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of x²ln4x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3