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Solving integrals/ (x^3+3x-2)÷(x+3)

Integral of (x^3+3x-2)÷(x+3) dx

The solution

You have entered [src]

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

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x^{3} + 3 x - 2}{x + 3} = x^{2} - 3 x + 12 - \frac{38}{x + 3}$
2. Integrate term-by-term:
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 x\right)\, dx = - 3 \int x\, dx$
    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: $- \frac{3 x^{2}}{2}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 12\, dx = 12 x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{38}{x + 3}\right)\, dx = - 38 \int \frac{1}{x + 3}\, dx$
    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: $- 38 \log{\left(x + 3 \right)}$
  The result is: $\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 12 x - 38 \log{\left(x + 3 \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x^{3} + 3 x - 2}{x + 3} = \frac{x^{3}}{x + 3} + \frac{3 x}{x + 3} - \frac{2}{x + 3}$
2. Integrate term-by-term:
  1. Rewrite the integrand:
    
    $\frac{x^{3}}{x + 3} = x^{2} - 3 x + 9 - \frac{27}{x + 3}$
  2. Integrate term-by-term:
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    1. 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: $- \frac{3 x^{2}}{2}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 9\, dx = 9 x$
    1. 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: $\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 9 x - 27 \log{\left(x + 3 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{3 x}{x + 3}\, dx = 3 \int \frac{x}{x + 3}\, dx$
    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: $3 x - 9 \log{\left(x + 3 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{2}{x + 3}\right)\, dx = - 2 \int \frac{1}{x + 3}\, dx$
    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: $- 2 \log{\left(x + 3 \right)}$
  The result is: $\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 12 x - 36 \log{\left(x + 3 \right)} - 2 \log{\left(x + 3 \right)}$
Add the constant of integration:

$\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 12 x - 38 \log{\left(x + 3 \right)}+ \mathrm{constant}$

The answer is:

$\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 12 x - 38 \log{\left(x + 3 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                      
 |                                                       
 |  3                                              2    3
 | x  + 3*x - 2                                 3*x    x 
 | ------------ dx = C - 38*log(3 + x) + 12*x - ---- + --
 |    x + 3                                      2     3 
 |                                                       
/

$${{2\,x^3-9\,x^2+72\,x}\over{6}}-38\,\log \left(x+3\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

65/6 - 38*log(4) + 38*log(3)

$$38\,\log 3-{{228\,\log 4-65}\over{6}}$$

65/6 - 38*log(4) + 38*log(3)

$$- 38 \log{\left(4 \right)} + \frac{65}{6} + 38 \log{\left(3 \right)}$$

Simplify

Numerical answer [src]

-0.0985854198343419

-0.0985854198343419

The graph

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

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Identical expressions

Similar expressions

Integral of (x^3+3x-2)÷(x+3) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2