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Identical expressions
(x^ two -x- one)/(x+ two)
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(x2-x-1)/(x+2)
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(x^2-x-1)/(x+2)dx
Similar expressions
(x^2-x-1)/(x-2)
(x^2-x+1)/(x+2)
(x^2+x-1)/(x+2)

Solving integrals/ (x^2-x-1)/(x+2)

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

The solution

You have entered [src]

  1              
  /              
 |               
 |   2           
 |  x  - x - 1   
 |  ---------- dx
 |    x + 2      
 |               
/                
0

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

Integral((x^2 - x - 1)/(x + 2), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = - x$ .
  
  Then let $du = - dx$ and substitute $du$ :
  
  $\int \frac{u^{2} + u - 1}{u - 2}\, du$
  1. Rewrite the integrand:
    
    $\frac{u^{2} + u - 1}{u - 2} = u + 3 + \frac{5}{u - 2}$
  2. Integrate term-by-term:
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 3\, du = 3 u$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{5}{u - 2}\, du = 5 \int \frac{1}{u - 2}\, du$
      
      Let $u = u - 2$ .
      
      Then let $du = du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(u - 2 \right)}$
      
      So, the result is: $5 \log{\left(u - 2 \right)}$
    The result is: $\frac{u^{2}}{2} + 3 u + 5 \log{\left(u - 2 \right)}$
  Now substitute $u$ back in:
  
  $\frac{x^{2}}{2} - 3 x + 5 \log{\left(- x - 2 \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{\left(x^{2} - x\right) - 1}{x + 2} = x - 3 + \frac{5}{x + 2}$
2. Integrate term-by-term:
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(-3\right)\, dx = - 3 x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{5}{x + 2}\, dx = 5 \int \frac{1}{x + 2}\, dx$
    1. Let $u = x + 2$ .
      
      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:
      
      $\log{\left(x + 2 \right)}$
    So, the result is: $5 \log{\left(x + 2 \right)}$
  The result is: $\frac{x^{2}}{2} - 3 x + 5 \log{\left(x + 2 \right)}$
Method #3
1. Rewrite the integrand:
  
  $\frac{\left(x^{2} - x\right) - 1}{x + 2} = \frac{x^{2}}{x + 2} - \frac{x}{x + 2} - \frac{1}{x + 2}$
2. Integrate term-by-term:
  1. Rewrite the integrand:
    
    $\frac{x^{2}}{x + 2} = x - 2 + \frac{4}{x + 2}$
  2. Integrate term-by-term:
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int \left(-2\right)\, dx = - 2 x$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{4}{x + 2}\, dx = 4 \int \frac{1}{x + 2}\, dx$
      
      Let $u = x + 2$ .
      
      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:
      
      $\log{\left(x + 2 \right)}$
      
      So, the result is: $4 \log{\left(x + 2 \right)}$
    The result is: $\frac{x^{2}}{2} - 2 x + 4 \log{\left(x + 2 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{x}{x + 2}\right)\, dx = - \int \frac{x}{x + 2}\, dx$
    1. Rewrite the integrand:
      
      $\frac{x}{x + 2} = 1 - \frac{2}{x + 2}$
    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:
      
      $\int \left(- \frac{2}{x + 2}\right)\, dx = - 2 \int \frac{1}{x + 2}\, dx$
      
      Let $u = x + 2$ .
      
      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:
      
      $\log{\left(x + 2 \right)}$
      
      So, the result is: $- 2 \log{\left(x + 2 \right)}$
      
      The result is: $x - 2 \log{\left(x + 2 \right)}$
    So, the result is: $- x + 2 \log{\left(x + 2 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{x + 2}\right)\, dx = - \int \frac{1}{x + 2}\, dx$
    1. Let $u = x + 2$ .
      
      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:
      
      $\log{\left(x + 2 \right)}$
    So, the result is: $- \log{\left(x + 2 \right)}$
  The result is: $\frac{x^{2}}{2} - 3 x - \log{\left(x + 2 \right)} + 6 \log{\left(x + 2 \right)}$
Add the constant of integration:

$\frac{x^{2}}{2} - 3 x + 5 \log{\left(- x - 2 \right)}+ \mathrm{constant}$

The answer is:

$\frac{x^{2}}{2} - 3 x + 5 \log{\left(- x - 2 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                            
 |                                             
 |  2                   2                      
 | x  - x - 1          x                       
 | ---------- dx = C + -- - 3*x + 5*log(-2 - x)
 |   x + 2             2                       
 |                                             
/

$$\int \frac{\left(x^{2} - x\right) - 1}{x + 2}\, dx = C + \frac{x^{2}}{2} - 3 x + 5 \log{\left(- x - 2 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-5/2 - 5*log(2) + 5*log(3)

$$- 5 \log{\left(2 \right)} - \frac{5}{2} + 5 \log{\left(3 \right)}$$

-5/2 - 5*log(2) + 5*log(3)

$$- 5 \log{\left(2 \right)} - \frac{5}{2} + 5 \log{\left(3 \right)}$$

-5/2 - 5*log(2) + 5*log(3)

Numerical answer [src]

-0.472674459459178

-0.472674459459178

The graph

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

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How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3