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Identical expressions
(x^ three - one)/(x+ three)
(x cubed minus 1) divide by (x plus 3)
(x to the power of three minus one) divide by (x plus three)
(x3-1)/(x+3)
x3-1/x+3
(x³-1)/(x+3)
(x to the power of 3-1)/(x+3)
x^3-1/x+3
(x^3-1) divide by (x+3)
(x^3-1)/(x+3)dx
Similar expressions
(x^3+1)/(x+3)
(x^3-1)/(x-3)

Solving integrals/ (x^3-1)/(x+3)

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

The solution

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

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x^{3} - 1}{x + 3} = x^{2} - 3 x + 9 - \frac{28}{x + 3}$
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:
    
    $\int 9\, dx = 9 x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{28}{x + 3}\right)\, dx = - 28 \int \frac{1}{x + 3}\, dx$
    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: $- 28 \log{\left(x + 3 \right)}$
  The result is: $\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 9 x - 28 \log{\left(x + 3 \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x^{3} - 1}{x + 3} = \frac{x^{3}}{x + 3} - \frac{1}{x + 3}$
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:
    
    $\int \left(- \frac{1}{x + 3}\right)\, dx = - \int \frac{1}{x + 3}\, dx$
    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: $- \log{\left(x + 3 \right)}$
  The result is: $\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 9 x - 27 \log{\left(x + 3 \right)} - \log{\left(x + 3 \right)}$
Add the constant of integration:

$\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 9 x - 28 \log{\left(x + 3 \right)}+ \mathrm{constant}$

The answer is:

\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 9 x - 28 \log{\left(x + 3 \right)}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                                               
 |                                                
 |  3                                       2    3
 | x  - 1                                3*x    x 
 | ------ dx = C - 28*log(3 + x) + 9*x - ---- + --
 | x + 3                                  2     3 
 |                                                
/

\int \frac{x^{3} - 1}{x + 3}\, dx = C + \frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 9 x - 28 \log{\left(x + 3 \right)}

Simplify

The graph

Plot the graph f(x)

The answer [src]

47/6 - 28*log(4) + 28*log(3)

- 28 \log{\left(4 \right)} + \frac{47}{6} + 28 \log{\left(3 \right)}

47/6 - 28*log(4) + 28*log(3)

- 28 \log{\left(4 \right)} + \frac{47}{6} + 28 \log{\left(3 \right)}

Упростить

Numerical answer [src]

-0.221764695316533

-0.221764695316533

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2