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Identical expressions
(two x- one)/(x+ three)(x^2- four)
(2x minus 1) divide by (x plus 3)(x squared minus 4)
(two x minus one) divide by (x plus three)(x squared minus four)
(2x-1)/(x+3)(x2-4)
2x-1/x+3x2-4
(2x-1)/(x+3)(x²-4)
(2x-1)/(x+3)(x to the power of 2-4)
2x-1/x+3x^2-4
(2x-1) divide by (x+3)(x^2-4)
(2x-1)/(x+3)(x^2-4)dx
Similar expressions
(2x-1)/(x+3)(x^2+4)
(2x+1)/(x+3)(x^2-4)
(2x-1)/(x-3)(x^2-4)

Integral of (2x-1)/(x+3)(x^2-4) dx

The solution

You have entered [src]

  1                    
  /                    
 |                     
 |  2*x - 1 / 2    \   
 |  -------*\x  - 4/ dx
 |   x + 3             
 |                     
/                      
0

$$\int\limits_{0}^{1} \frac{2 x - 1}{x + 3} \left(x^{2} - 4\right)\, dx$$

Integral(((2*x - 1)/(x + 3))*(x^2 - 4), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{2 x - 1}{x + 3} \left(x^{2} - 4\right) = 2 x^{2} - 7 x + 13 - \frac{35}{x + 3}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 x^{2}\, dx = 2 \int x^{2}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    So, the result is: $\frac{2 x^{3}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 7 x\right)\, dx = - 7 \int x\, dx$
    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: $- \frac{7 x^{2}}{2}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 13\, dx = 13 x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{35}{x + 3}\right)\, dx = - 35 \int \frac{1}{x + 3}\, dx$
    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: $- 35 \log{\left(x + 3 \right)}$
  The result is: $\frac{2 x^{3}}{3} - \frac{7 x^{2}}{2} + 13 x - 35 \log{\left(x + 3 \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{2 x - 1}{x + 3} \left(x^{2} - 4\right) = \frac{2 x^{3} - x^{2} - 8 x + 4}{x + 3}$
2. Rewrite the integrand:
  
  $\frac{2 x^{3} - x^{2} - 8 x + 4}{x + 3} = 2 x^{2} - 7 x + 13 - \frac{35}{x + 3}$
3. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 x^{2}\, dx = 2 \int x^{2}\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    So, the result is: $\frac{2 x^{3}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 7 x\right)\, dx = - 7 \int x\, dx$
    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: $- \frac{7 x^{2}}{2}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 13\, dx = 13 x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{35}{x + 3}\right)\, dx = - 35 \int \frac{1}{x + 3}\, dx$
    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: $- 35 \log{\left(x + 3 \right)}$
  The result is: $\frac{2 x^{3}}{3} - \frac{7 x^{2}}{2} + 13 x - 35 \log{\left(x + 3 \right)}$
Method #3
1. Rewrite the integrand:
  
  $\frac{2 x - 1}{x + 3} \left(x^{2} - 4\right) = \frac{2 x^{3}}{x + 3} - \frac{x^{2}}{x + 3} - \frac{8 x}{x + 3} + \frac{4}{x + 3}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2 x^{3}}{x + 3}\, dx = 2 \int \frac{x^{3}}{x + 3}\, dx$
    1. Rewrite the integrand:
      
      $\frac{x^{3}}{x + 3} = x^{2} - 3 x + 9 - \frac{27}{x + 3}$
    2. Integrate term-by-term:
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 3 x\right)\, dx = - 3 \int x\, dx$
      
      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: $- \frac{3 x^{2}}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 9\, dx = 9 x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{27}{x + 3}\right)\, dx = - 27 \int \frac{1}{x + 3}\, dx$
      
      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: $- 27 \log{\left(x + 3 \right)}$
      
      The result is: $\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 9 x - 27 \log{\left(x + 3 \right)}$
    So, the result is: $\frac{2 x^{3}}{3} - 3 x^{2} + 18 x - 54 \log{\left(x + 3 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{x^{2}}{x + 3}\right)\, dx = - \int \frac{x^{2}}{x + 3}\, dx$
    1. Rewrite the integrand:
      
      $\frac{x^{2}}{x + 3} = x - 3 + \frac{9}{x + 3}$
    2. Integrate term-by-term:
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \left(-3\right)\, dx = - 3 x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{9}{x + 3}\, dx = 9 \int \frac{1}{x + 3}\, dx$
      
      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: $9 \log{\left(x + 3 \right)}$
      
      The result is: $\frac{x^{2}}{2} - 3 x + 9 \log{\left(x + 3 \right)}$
    So, the result is: $- \frac{x^{2}}{2} + 3 x - 9 \log{\left(x + 3 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{8 x}{x + 3}\right)\, dx = - 8 \int \frac{x}{x + 3}\, dx$
    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: $- 8 x + 24 \log{\left(x + 3 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{4}{x + 3}\, dx = 4 \int \frac{1}{x + 3}\, dx$
    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: $4 \log{\left(x + 3 \right)}$
  The result is: $\frac{2 x^{3}}{3} - \frac{7 x^{2}}{2} + 13 x + 4 \log{\left(x + 3 \right)} - 39 \log{\left(x + 3 \right)}$
Add the constant of integration:

$\frac{2 x^{3}}{3} - \frac{7 x^{2}}{2} + 13 x - 35 \log{\left(x + 3 \right)}+ \mathrm{constant}$

The answer is:

$\frac{2 x^{3}}{3} - \frac{7 x^{2}}{2} + 13 x - 35 \log{\left(x + 3 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                            
 |                                                     2      3
 | 2*x - 1 / 2    \                                 7*x    2*x 
 | -------*\x  - 4/ dx = C - 35*log(3 + x) + 13*x - ---- + ----
 |  x + 3                                            2      3  
 |                                                             
/

$$\int \frac{2 x - 1}{x + 3} \left(x^{2} - 4\right)\, dx = C + \frac{2 x^{3}}{3} - \frac{7 x^{2}}{2} + 13 x - 35 \log{\left(x + 3 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

61/6 - 35*log(4) + 35*log(3)

$$- 35 \log{\left(4 \right)} + \frac{61}{6} + 35 \log{\left(3 \right)}$$

61/6 - 35*log(4) + 35*log(3)

$$- 35 \log{\left(4 \right)} + \frac{61}{6} + 35 \log{\left(3 \right)}$$

61/6 - 35*log(4) + 35*log(3)

Numerical answer [src]

0.0977941308543342

0.0977941308543342

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 (2x-1)/(x+3)(x^2-4) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3