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Integral of xcos(5x)
Identical expressions
x/ five *lg(x^ two / five)
x divide by 5 multiply by lg(x squared divide by 5)
x divide by five multiply by lg(x to the power of two divide by five)
x/5*lg(x2/5)
x/5*lgx2/5
x/5*lg(x²/5)
x/5*lg(x to the power of 2/5)
x/5lg(x^2/5)
x/5lg(x2/5)
x/5lgx2/5
x/5lgx^2/5
x divide by 5*lg(x^2 divide by 5)
x/5*lg(x^2/5)dx

Integral of x/5*lg(x^2/5) dx

The solution

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  4             
  /             
 |              
 |       / 2\   
 |  x    |x |   
 |  -*log|--| dx
 |  5    \5 /   
 |              
/               
1

$$\int\limits_{1}^{4} \frac{x}{5} \log{\left(\frac{x^{2}}{5} \right)}\, dx$$

Integral((x/5)*log(x^2/5), (x, 1, 4))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \frac{x^{2}}{5}$ .
  
  Then let $du = \frac{2 x dx}{5}$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{\log{\left(u \right)}}{2}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \log{\left(u \right)}\, du = \frac{\int \log{\left(u \right)}\, du}{2}$
    1. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(u \right)} = \log{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = 1$ .
      
      Then $\operatorname{du}{\left(u \right)} = \frac{1}{u}$ .
      
      To find $v{\left(u \right)}$ :
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      Now evaluate the sub-integral.
    2. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
    So, the result is: $\frac{u \log{\left(u \right)}}{2} - \frac{u}{2}$
  Now substitute $u$ back in:
  
  $\frac{x^{2} \log{\left(\frac{x^{2}}{5} \right)}}{10} - \frac{x^{2}}{10}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x}{5} \log{\left(\frac{x^{2}}{5} \right)} = \frac{x \log{\left(x^{2} \right)}}{5} - \frac{x \log{\left(5 \right)}}{5}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{x \log{\left(x^{2} \right)}}{5}\, dx = \frac{\int x \log{\left(x^{2} \right)}\, dx}{5}$
    1. Let $u = \log{\left(x^{2} \right)}$ .
      
      Then let $du = \frac{2 dx}{x}$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{u e^{u}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u e^{u}\, du = \frac{\int u e^{u}\, du}{2}$
      
      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 $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\operatorname{du}{\left(u \right)} = 1$ .
      
      To find $v{\left(u \right)}$ :
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      Now evaluate the sub-integral.
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{u e^{u}}{2} - \frac{e^{u}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{x^{2} \log{\left(x^{2} \right)}}{2} - \frac{x^{2}}{2}$
    So, the result is: $\frac{x^{2} \log{\left(x^{2} \right)}}{10} - \frac{x^{2}}{10}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{x \log{\left(5 \right)}}{5}\right)\, dx = - \frac{\log{\left(5 \right)} \int x\, dx}{5}$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
    So, the result is: $- \frac{x^{2} \log{\left(5 \right)}}{10}$
  The result is: $\frac{x^{2} \log{\left(x^{2} \right)}}{10} - \frac{x^{2} \log{\left(5 \right)}}{10} - \frac{x^{2}}{10}$
Method #3
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \log{\left(\frac{x^{2}}{5} \right)}$ and let $\operatorname{dv}{\left(x \right)} = \frac{x}{5}$ .
  
  Then $\operatorname{du}{\left(x \right)} = \frac{2}{x}$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{x}{5}\, dx = \frac{\int x\, dx}{5}$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
    So, the result is: $\frac{x^{2}}{10}$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x}{5}\, dx = \frac{\int x\, dx}{5}$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  So, the result is: $\frac{x^{2}}{10}$
Method #4
1. Rewrite the integrand:
  
  $\frac{x}{5} \log{\left(\frac{x^{2}}{5} \right)} = \frac{x \log{\left(x^{2} \right)}}{5} - \frac{x \log{\left(5 \right)}}{5}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{x \log{\left(x^{2} \right)}}{5}\, dx = \frac{\int x \log{\left(x^{2} \right)}\, dx}{5}$
    1. Let $u = \log{\left(x^{2} \right)}$ .
      
      Then let $du = \frac{2 dx}{x}$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{u e^{u}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u e^{u}\, du = \frac{\int u e^{u}\, du}{2}$
      
      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 $\operatorname{dv}{\left(u \right)} = e^{u}$ .
      
      Then $\operatorname{du}{\left(u \right)} = 1$ .
      
      To find $v{\left(u \right)}$ :
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      Now evaluate the sub-integral.
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{u e^{u}}{2} - \frac{e^{u}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{x^{2} \log{\left(x^{2} \right)}}{2} - \frac{x^{2}}{2}$
    So, the result is: $\frac{x^{2} \log{\left(x^{2} \right)}}{10} - \frac{x^{2}}{10}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{x \log{\left(5 \right)}}{5}\right)\, dx = - \frac{\log{\left(5 \right)} \int x\, dx}{5}$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
    So, the result is: $- \frac{x^{2} \log{\left(5 \right)}}{10}$
  The result is: $\frac{x^{2} \log{\left(x^{2} \right)}}{10} - \frac{x^{2} \log{\left(5 \right)}}{10} - \frac{x^{2}}{10}$
Now simplify:

$\frac{x^{2} \left(\log{\left(\frac{x^{2}}{5} \right)} - 1\right)}{10}$
Add the constant of integration:

$\frac{x^{2} \left(\log{\left(\frac{x^{2}}{5} \right)} - 1\right)}{10}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(\log{\left(\frac{x^{2}}{5} \right)} - 1\right)}{10}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                              / 2\
 |                          2    |x |
 |      / 2\           2   x *log|--|
 | x    |x |          x          \5 /
 | -*log|--| dx = C - -- + ----------
 | 5    \5 /          10       10    
 |                                   
/

$$\int \frac{x}{5} \log{\left(\frac{x^{2}}{5} \right)}\, dx = C + \frac{x^{2} \log{\left(\frac{x^{2}}{5} \right)}}{10} - \frac{x^{2}}{10}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  3   log(5)   8*log(16/5)
- - + ------ + -----------
  2     10          5

$$- \frac{3}{2} + \frac{\log{\left(5 \right)}}{10} + \frac{8 \log{\left(\frac{16}{5} \right)}}{5}$$

  3   log(5)   8*log(16/5)
- - + ------ + -----------
  2     10          5

$$- \frac{3}{2} + \frac{\log{\left(5 \right)}}{10} + \frac{8 \log{\left(\frac{16}{5} \right)}}{5}$$

-3/2 + log(5)/10 + 8*log(16/5)/5

Numerical answer [src]

0.521985086932499

0.521985086932499

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

Other calculators

How to use it?

Identical expressions

Integral of x/5*lg(x^2/5) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3

Method #4