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Derivative of:
(ln(x+3))/(x+3)
Identical expressions
(ln(x+ three))/(x+ three)
(ln(x plus 3)) divide by (x plus 3)
(ln(x plus three)) divide by (x plus three)
lnx+3/x+3
(ln(x+3)) divide by (x+3)
(ln(x+3))/(x+3)dx
Similar expressions
(ln(x+3))/(x-3)
(ln(x-3))/(x+3)

Integral of (ln(x+3))/(x+3) dx

The solution

You have entered [src]

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

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

   2         2   
log (4)   log (3)
------- - -------
   2         2

$$- \frac{\log{\left(3 \right)}^{2}}{2} + \frac{\log{\left(4 \right)}^{2}}{2}$$

   2         2   
log (4)   log (3)
------- - -------
   2         2

$$- \frac{\log{\left(3 \right)}^{2}}{2} + \frac{\log{\left(4 \right)}^{2}}{2}$$

log(4)^2/2 - log(3)^2/2

Numerical answer [src]

0.357431547430112

0.357431547430112

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

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Identical expressions

Similar expressions

Integral of (ln(x+3))/(x+3) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2