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

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

$\frac{x^{3}}{x + 1} = x^{2} - x + 1 - \frac{1}{x + 1}$
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:
  
  $\int \left(- x\right)\, dx = - \int x\, dx$
  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:
  
  $\int 1\, dx = x$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{x + 1}\right)\, dx = - \int \frac{1}{x + 1}\, dx$
  1. Let $u = x + 1$ .
    
    Then let $du = dx$ and substitute $du$ :
    
    $\int \frac{1}{u}\, du$
    1. 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: $- \log{\left(x + 1 \right)}$
The result is: $\frac{x^{3}}{3} - \frac{x^{2}}{2} + x - \log{\left(x + 1 \right)}$
Add the constant of integration:

$\frac{x^{3}}{3} - \frac{x^{2}}{2} + x - \log{\left(x + 1 \right)}+ \mathrm{constant}$

The answer is:

$\frac{x^{3}}{3} - \frac{x^{2}}{2} + x - \log{\left(x + 1 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

5/6 - log(2)

$$\frac{5}{6} - \log{\left(2 \right)}$$

5/6 - log(2)

$$\frac{5}{6} - \log{\left(2 \right)}$$

5/6 - log(2)

Numerical answer [src]

0.140186152773388

0.140186152773388

The graph

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution