Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of 1/(2+x^2)
Integral of x*sin2x
Integral of x/(x^3-3x+2)
Integral of -x+4
Graphing y =:
(2x-1)/(x+3)
Identical expressions
(2x- one)/(x+ three)
(2x minus 1) divide by (x plus 3)
(2x minus one) divide by (x plus three)
2x-1/x+3
(2x-1) divide by (x+3)
(2x-1)/(x+3)dx
Similar expressions
(2x-1)/((x+3)(x+1))
(2x+1)/(x+3)
(2x-1)/(x-3)

Integral of (2x-1)/(x+3) dx

The solution

You have entered [src]

  3           
  /           
 |            
 |  2*x - 1   
 |  ------- dx
 |   x + 3    
 |            
/             
2

$$\int\limits_{2}^{3} \frac{2 x - 1}{x + 3}\, dx$$

Integral((2*x - 1)/(x + 3), (x, 2, 3))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 2 x$ .
  
  Then let $du = 2 dx$ and substitute $du$ :
  
  $\int \frac{u - 1}{u + 6}\, du$
  1. Let $u = u + 6$ .
    
    Then let $du = du$ and substitute $du$ :
    
    $\int \frac{u - 7}{u}\, du$
    1. Rewrite the integrand:
      
      $\frac{u - 7}{u} = 1 - \frac{7}{u}$
    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 a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{7}{u}\right)\, du = - 7 \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- 7 \log{\left(u \right)}$
      
      The result is: $u - 7 \log{\left(u \right)}$
    Now substitute $u$ back in:
    
    $u - 7 \log{\left(u + 6 \right)} + 6$
  Now substitute $u$ back in:
  
  $2 x - 7 \log{\left(2 x + 6 \right)} + 6$
Method #2
1. Rewrite the integrand:
  
  $\frac{2 x - 1}{x + 3} = 2 - \frac{7}{x + 3}$
2. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 2\, dx = 2 x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{7}{x + 3}\right)\, dx = - 7 \int \frac{1}{x + 3}\, dx$
    1. Let $u = x + 3$ .
      
      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:
      
      $\log{\left(x + 3 \right)}$
    So, the result is: $- 7 \log{\left(x + 3 \right)}$
  The result is: $2 x - 7 \log{\left(x + 3 \right)}$
Method #3
1. Rewrite the integrand:
  
  $\frac{2 x - 1}{x + 3} = \frac{2 x}{x + 3} - \frac{1}{x + 3}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2 x}{x + 3}\, dx = 2 \int \frac{x}{x + 3}\, dx$
    1. Rewrite the integrand:
      
      $\frac{x}{x + 3} = 1 - \frac{3}{x + 3}$
    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:
      
      $\int \left(- \frac{3}{x + 3}\right)\, dx = - 3 \int \frac{1}{x + 3}\, dx$
      
      Let $u = x + 3$ .
      
      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:
      
      $\log{\left(x + 3 \right)}$
      
      So, the result is: $- 3 \log{\left(x + 3 \right)}$
      
      The result is: $x - 3 \log{\left(x + 3 \right)}$
    So, the result is: $2 x - 6 \log{\left(x + 3 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{x + 3}\right)\, dx = - \int \frac{1}{x + 3}\, dx$
    1. Let $u = x + 3$ .
      
      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:
      
      $\log{\left(x + 3 \right)}$
    So, the result is: $- \log{\left(x + 3 \right)}$
  The result is: $2 x - 6 \log{\left(x + 3 \right)} - \log{\left(x + 3 \right)}$
Add the constant of integration:

$2 x - 7 \log{\left(2 x + 6 \right)} + 6+ \mathrm{constant}$

The answer is:

$2 x - 7 \log{\left(2 x + 6 \right)} + 6+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
 |                                          
 | 2*x - 1                                  
 | ------- dx = 6 + C - 7*log(6 + 2*x) + 2*x
 |  x + 3                                   
 |                                          
/

$$\int \frac{2 x - 1}{x + 3}\, dx = C + 2 x - 7 \log{\left(2 x + 6 \right)} + 6$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

2 - 7*log(6) + 7*log(5)

$$- 7 \log{\left(6 \right)} + 2 + 7 \log{\left(5 \right)}$$

2 - 7*log(6) + 7*log(5)

$$- 7 \log{\left(6 \right)} + 2 + 7 \log{\left(5 \right)}$$

2 - 7*log(6) + 7*log(5)

Numerical answer [src]

0.723749102442318

0.723749102442318

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (2x-1)/(x+3) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3