Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of 1/(2+x^2)
Integral of sin(w*t)^2
Integral of sin(log(x))^2/x
Integral of sin^n(x)
Identical expressions
(ln(two x+2))/(x+ one)
(ln(2x plus 2)) divide by (x plus 1)
(ln(two x plus 2)) divide by (x plus one)
ln2x+2/x+1
(ln(2x+2)) divide by (x+1)
(ln(2x+2))/(x+1)dx
Similar expressions
(ln(2x-2))/(x+1)
(ln(2x+2))/(x-1)

Solving integrals/ (ln(2x+2))/(x+1)

Integral of (ln(2x+2))/(x+1) dx

The solution

You have entered [src]

 oo                
  /                
 |                 
 |  log(2*x + 2)   
 |  ------------ dx
 |     x + 1       
 |                 
/                  
0

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

Integral(log(2*x + 2)/(x + 1), (x, 0, oo))

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                   
 |                          2         
 | log(2*x + 2)          log (2*x + 2)
 | ------------ dx = C + -------------
 |    x + 1                    2      
 |                                    
/

$$\int \frac{\log{\left(2 x + 2 \right)}}{x + 1}\, dx = C + \frac{\log{\left(2 x + 2 \right)}^{2}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

Упростить

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (ln(2x+2))/(x+1) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2