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Integral of d{x}:
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Graphing y =:
(x^2-3)/(2-x)
Identical expressions
(x^ two - three)/(two -x)
(x squared minus 3) divide by (2 minus x)
(x to the power of two minus three) divide by (two minus x)
(x2-3)/(2-x)
x2-3/2-x
(x²-3)/(2-x)
(x to the power of 2-3)/(2-x)
x^2-3/2-x
(x^2-3) divide by (2-x)
(x^2-3)/(2-x)dx
Similar expressions
(x^2+3)/(2-x)
(x^2-3)/(2+x)

Integral of (x^2-3)/(2-x) dx

The solution

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  1          
  /          
 |           
 |   2       
 |  x  - 3   
 |  ------ dx
 |  2 - x    
 |           
/            
0

$$\int\limits_{0}^{1} \frac{x^{2} - 3}{2 - x}\, dx$$

Integral((x^2 - 3)/(2 - x), (x, 0, 1))

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-5/2 + log(2)

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

-5/2 + log(2)

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

-5/2 + log(2)

Numerical answer [src]

-1.80685281944005

-1.80685281944005

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (x^2-3)/(2-x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3

Method #1

Method #2

Method #3