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Identical expressions
(3x+ four)/(x+ one)
(3x plus 4) divide by (x plus 1)
(3x plus four) divide by (x plus one)
3x+4/x+1
(3x+4) divide by (x+1)
(3x+4)/(x+1)dx
Similar expressions
(3x-4)/(x+1)
(3x+4)/(x-1)

Integral of (3x+4)/(x+1) dx

The solution

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  1           
  /           
 |            
 |  3*x + 4   
 |  ------- dx
 |   x + 1    
 |            
/             
0

\int\limits_{0}^{1} \frac{3 x + 4}{x + 1}\, dx

Integral((3*x + 4)/(x + 1), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $du$ :
  
  $\int \frac{u + 4}{u + 3}\, du$
  1. Let $u = u + 3$ .
    
    Then let $du = du$ and substitute $du$ :
    
    $\int \frac{u + 1}{u}\, du$
    1. Rewrite the integrand:
      
      $\frac{u + 1}{u} = 1 + \frac{1}{u}$
    2. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      The result is: $u + \log{\left(u \right)}$
    Now substitute $u$ back in:
    
    $u + \log{\left(u + 3 \right)} + 3$
  Now substitute $u$ back in:
  
  $3 x + \log{\left(3 x + 3 \right)} + 3$
Method #2
1. Rewrite the integrand:
  
  $\frac{3 x + 4}{x + 1} = 3 + \frac{1}{x + 1}$
2. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 3\, dx = 3 x$
  1. Let $u = x + 1$ .
    
    Then let $du = dx$ and substitute $du$ :
    
    $\int \frac{1}{u}\, du$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    Now substitute $u$ back in:
    
    $\log{\left(x + 1 \right)}$
  The result is: $3 x + \log{\left(x + 1 \right)}$
Method #3
1. Rewrite the integrand:
  
  $\frac{3 x + 4}{x + 1} = \frac{3 x}{x + 1} + \frac{4}{x + 1}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{3 x}{x + 1}\, dx = 3 \int \frac{x}{x + 1}\, dx$
    1. Rewrite the integrand:
      
      $\frac{x}{x + 1} = 1 - \frac{1}{x + 1}$
    2. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x + 1}\right)\, dx = - \int \frac{1}{x + 1}\, dx$
      
      Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 1 \right)}$
      
      So, the result is: $- \log{\left(x + 1 \right)}$
      
      The result is: $x - \log{\left(x + 1 \right)}$
    So, the result is: $3 x - 3 \log{\left(x + 1 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{4}{x + 1}\, dx = 4 \int \frac{1}{x + 1}\, dx$
    1. Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 1 \right)}$
    So, the result is: $4 \log{\left(x + 1 \right)}$
  The result is: $3 x - 3 \log{\left(x + 1 \right)} + 4 \log{\left(x + 1 \right)}$
Add the constant of integration:

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

The answer is:

3 x + \log{\left(3 x + 3 \right)} + 3+ \mathrm{constant}

The answer (Indefinite) [src]

  /                                       
 |                                        
 | 3*x + 4                                
 | ------- dx = 3 + C + 3*x + log(3 + 3*x)
 |  x + 1                                 
 |                                        
/

\int \frac{3 x + 4}{x + 1}\, dx = C + 3 x + \log{\left(3 x + 3 \right)} + 3

Simplify

The graph

Plot the graph f(x)

The answer [src]

3 + log(2)

\log{\left(2 \right)} + 3

3 + log(2)

\log{\left(2 \right)} + 3

3 + log(2)

Numerical answer [src]

3.69314718055995

3.69314718055995

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

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Identical expressions

Similar expressions

Integral of (3x+4)/(x+1) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3