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Integral of d{x}:
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Identical expressions
ln^ two (x- three)/(x- three)
ln squared (x minus 3) divide by (x minus 3)
ln to the power of two (x minus three) divide by (x minus three)
ln2(x-3)/(x-3)
ln2x-3/x-3
ln²(x-3)/(x-3)
ln to the power of 2(x-3)/(x-3)
ln^2x-3/x-3
ln^2(x-3) divide by (x-3)
ln^2(x-3)/(x-3)dx
Similar expressions
ln^2(x+3)/(x-3)
ln^2(x-3)/(x+3)

Integral of ln^2(x-3)/(x-3) dx

The solution

You have entered [src]

  1               
  /               
 |                
 |     2          
 |  log (x - 3)   
 |  ----------- dx
 |     x - 3      
 |                
/                 
0

$$\int\limits_{0}^{1} \frac{\log{\left(x - 3 \right)}^{2}}{x - 3}\, dx$$

Integral(log(x - 3)^2/(x - 3), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \log{\left(x - 3 \right)}$ .
  
  Then let $du = \frac{dx}{x - 3}$ and substitute $du$ :
  
  $\int u^{2}\, du$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int u^{2}\, du = \frac{u^{3}}{3}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(x - 3 \right)}^{3}}{3}$
Method #2
1. Let $u = x - 3$ .
  
  Then let $du = dx$ and substitute $du$ :
  
  $\int \frac{\log{\left(u \right)}^{2}}{u}\, du$
  1. Let $u = \frac{1}{u}$ .
    
    Then let $du = - \frac{du}{u^{2}}$ and substitute $- du$ :
    
    $\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\, du$
      
      Let $u = \log{\left(\frac{1}{u} \right)}$ .
      
      Then let $du = - \frac{du}{u}$ and substitute $- du$ :
      
      $\int \left(- u^{2}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{2}\, du = - \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $- \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $- \frac{\log{\left(\frac{1}{u} \right)}^{3}}{3}$
      
      So, the result is: $\frac{\log{\left(\frac{1}{u} \right)}^{3}}{3}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(u \right)}^{3}}{3}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(x - 3 \right)}^{3}}{3}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                 3                  3
  (pi*I + log(3))    (pi*I + log(2)) 
- ---------------- + ----------------
         3                  3

$$\frac{\left(\log{\left(2 \right)} + i \pi\right)^{3}}{3} - \frac{\left(\log{\left(3 \right)} + i \pi\right)^{3}}{3}$$

                 3                  3
  (pi*I + log(3))    (pi*I + log(2)) 
- ---------------- + ----------------
         3                  3

$$\frac{\left(\log{\left(2 \right)} + i \pi\right)^{3}}{3} - \frac{\left(\log{\left(3 \right)} + i \pi\right)^{3}}{3}$$

-(pi*i + log(3))^3/3 + (pi*i + log(2))^3/3

Numerical answer [src]

(3.67079877942085 - 2.28235432962615j)

(3.67079877942085 - 2.28235432962615j)

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of ln^2(x-3)/(x-3) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2