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Integral of d{x}:
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Integral of sqrt(x^2+1)/x^3
Derivative of:
(x^2-3)/(x+1)
Graphing y =:
(x^2-3)/(x+1)
Identical expressions
(x^ two - three)/(x+ one)
(x squared minus 3) divide by (x plus 1)
(x to the power of two minus three) divide by (x plus one)
(x2-3)/(x+1)
x2-3/x+1
(x²-3)/(x+1)
(x to the power of 2-3)/(x+1)
x^2-3/x+1
(x^2-3) divide by (x+1)
(x^2-3)/(x+1)dx
Similar expressions
(x^2-3)/(x-1)
(x^2+3)/(x+1)

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

The solution

You have entered [src]

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

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x^{2} - 3}{x + 1} = x - 1 - \frac{2}{x + 1}$
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:
    
    $\int \left(-1\right)\, dx = - x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{2}{x + 1}\right)\, dx = - 2 \int \frac{1}{x + 1}\, dx$
    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: $- 2 \log{\left(x + 1 \right)}$
  The result is: $\frac{x^{2}}{2} - x - 2 \log{\left(x + 1 \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x^{2} - 3}{x + 1} = \frac{x^{2}}{x + 1} - \frac{3}{x + 1}$
2. Integrate term-by-term:
  1. Rewrite the integrand:
    
    $\frac{x^{2}}{x + 1} = x - 1 + \frac{1}{x + 1}$
  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:
      
      $\int \left(-1\right)\, dx = - x$
    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:
      
      $\log{\left(x + 1 \right)}$
    The result is: $\frac{x^{2}}{2} - x + \log{\left(x + 1 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{3}{x + 1}\right)\, dx = - 3 \int \frac{1}{x + 1}\, dx$
    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: $- 3 \log{\left(x + 1 \right)}$
  The result is: $\frac{x^{2}}{2} - x + \log{\left(x + 1 \right)} - 3 \log{\left(x + 1 \right)}$
Add the constant of integration:

$\frac{x^{2}}{2} - x - 2 \log{\left(x + 1 \right)}+ \mathrm{constant}$

The answer is:

$\frac{x^{2}}{2} - x - 2 \log{\left(x + 1 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/2 - 2*log(3) + 2*log(2)

$$- 2 \log{\left(3 \right)} + \frac{1}{2} + 2 \log{\left(2 \right)}$$

1/2 - 2*log(3) + 2*log(2)

$$- 2 \log{\left(3 \right)} + \frac{1}{2} + 2 \log{\left(2 \right)}$$

1/2 - 2*log(3) + 2*log(2)

Numerical answer [src]

-0.310930216216329

-0.310930216216329

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)/(x+1) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2