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Integral of d{x}:
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Identical expressions
x^ eight *ln(5x)
x to the power of 8 multiply by ln(5x)
x to the power of eight multiply by ln(5x)
x8*ln(5x)
x8*ln5x
x⁸*ln(5x)
x^8ln(5x)
x8ln(5x)
x8ln5x
x^8ln5x
x^8*ln(5x)dx

Integral of x^8*ln(5x) dx

The solution

You have entered [src]

  1               
  /               
 |                
 |   8            
 |  x *log(5*x) dx
 |                
/                 
0

$$\int\limits_{0}^{1} x^{8} \log{\left(5 x \right)}\, dx$$

Integral(x^8*log(5*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $x^{8} \log{\left(5 x \right)} = x^{8} \log{\left(x \right)} + x^{8} \log{\left(5 \right)}$
2. Integrate term-by-term:
  1. Let $u = \log{\left(x \right)}$ .
    
    Then let $du = \frac{dx}{x}$ and substitute $du$ :
    
    $\int u e^{9 u}\, du$
    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 $\operatorname{dv}{\left(u \right)} = e^{9 u}$ .
      
      Then $\operatorname{du}{\left(u \right)} = 1$ .
      
      To find $v{\left(u \right)}$ :
      
      There are multiple ways to do this integral.
      
      Method #1
      
      Let $u = 9 u$ .
      
      Then let $du = 9 du$ and substitute $\frac{du}{9}$ :
      
      $\int \frac{e^{u}}{81}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{9}\, du = \frac{\int e^{u}\, du}{9}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{9}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{9 u}}{9}$
      
      Method #2
      
      Let $u = e^{9 u}$ .
      
      Then let $du = 9 e^{9 u} du$ and substitute $\frac{du}{9}$ :
      
      $\int \frac{1}{81}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{9}\, du = \frac{\int 1\, du}{9}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{u}{9}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{9 u}}{9}$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{9 u}}{9}\, du = \frac{\int e^{9 u}\, du}{9}$
      
      Let $u = 9 u$ .
      
      Then let $du = 9 du$ and substitute $\frac{du}{9}$ :
      
      $\int \frac{e^{u}}{81}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{9}\, du = \frac{\int e^{u}\, du}{9}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{9}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{9 u}}{9}$
      
      So, the result is: $\frac{e^{9 u}}{81}$
    Now substitute $u$ back in:
    
    $\frac{x^{9} \log{\left(x \right)}}{9} - \frac{x^{9}}{81}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int x^{8} \log{\left(5 \right)}\, dx = \log{\left(5 \right)} \int x^{8}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{8}\, dx = \frac{x^{9}}{9}$
    So, the result is: $\frac{x^{9} \log{\left(5 \right)}}{9}$
  The result is: $\frac{x^{9} \log{\left(x \right)}}{9} - \frac{x^{9}}{81} + \frac{x^{9} \log{\left(5 \right)}}{9}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \log{\left(5 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = x^{8}$ .
  
  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 $n \neq -1$ :
    
    $\int x^{8}\, dx = \frac{x^{9}}{9}$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x^{8}}{9}\, dx = \frac{\int x^{8}\, dx}{9}$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{8}\, dx = \frac{x^{9}}{9}$
  So, the result is: $\frac{x^{9}}{81}$
Method #3
1. Rewrite the integrand:
  
  $x^{8} \log{\left(5 x \right)} = x^{8} \log{\left(x \right)} + x^{8} \log{\left(5 \right)}$
2. Integrate term-by-term:
  1. Let $u = \log{\left(x \right)}$ .
    
    Then let $du = \frac{dx}{x}$ and substitute $du$ :
    
    $\int u e^{9 u}\, du$
    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 $\operatorname{dv}{\left(u \right)} = e^{9 u}$ .
      
      Then $\operatorname{du}{\left(u \right)} = 1$ .
      
      To find $v{\left(u \right)}$ :
      
      Let $u = 9 u$ .
      
      Then let $du = 9 du$ and substitute $\frac{du}{9}$ :
      
      $\int \frac{e^{u}}{81}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{9}\, du = \frac{\int e^{u}\, du}{9}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{9}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{9 u}}{9}$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{9 u}}{9}\, du = \frac{\int e^{9 u}\, du}{9}$
      
      Let $u = 9 u$ .
      
      Then let $du = 9 du$ and substitute $\frac{du}{9}$ :
      
      $\int \frac{e^{u}}{81}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{9}\, du = \frac{\int e^{u}\, du}{9}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{9}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{9 u}}{9}$
      
      So, the result is: $\frac{e^{9 u}}{81}$
    Now substitute $u$ back in:
    
    $\frac{x^{9} \log{\left(x \right)}}{9} - \frac{x^{9}}{81}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int x^{8} \log{\left(5 \right)}\, dx = \log{\left(5 \right)} \int x^{8}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{8}\, dx = \frac{x^{9}}{9}$
    So, the result is: $\frac{x^{9} \log{\left(5 \right)}}{9}$
  The result is: $\frac{x^{9} \log{\left(x \right)}}{9} - \frac{x^{9}}{81} + \frac{x^{9} \log{\left(5 \right)}}{9}$
Now simplify:

$\frac{x^{9} \cdot \left(9 \log{\left(x \right)} - 1 + \log{\left(1953125 \right)}\right)}{81}$
Add the constant of integration:

$\frac{x^{9} \cdot \left(9 \log{\left(x \right)} - 1 + \log{\left(1953125 \right)}\right)}{81}+ \mathrm{constant}$

The answer is:

$\frac{x^{9} \cdot \left(9 \log{\left(x \right)} - 1 + \log{\left(1953125 \right)}\right)}{81}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                               
 |                       9    9           9       
 |  8                   x    x *log(5)   x *log(x)
 | x *log(5*x) dx = C - -- + --------- + ---------
 |                      81       9           9    
/

$${{x^9\,\log \left(5\,x\right)}\over{9}}-{{x^9}\over{81}}$$

Simplify

The answer [src]

  1    log(5)
- -- + ------
  81     9

$${{17578125\,\log 5-1953125}\over{158203125}}$$

  1    log(5)
- -- + ------
  81     9

$$- \frac{1}{81} + \frac{\log{\left(5 \right)}}{9}$$

Simplify

Numerical answer [src]

0.166480755702554

0.166480755702554

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

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How to use it?

Identical expressions

Integral of x^8*ln(5x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #1

Method #2

Method #2

Method #3