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(x^ three)/(x+ three)
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Similar expressions
(x^3)/(x-3)

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

$\frac{x^{3}}{x + 3} = x^{2} - 3 x + 9 - \frac{27}{x + 3}$
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:
  
  $\int \left(- \frac{27}{x + 3}\right)\, dx = - 27 \int \frac{1}{x + 3}\, dx$
  1. Let $u = x + 3$ .
    
    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:
    
    $\log{\left(x + 3 \right)}$
  So, the result is: $- 27 \log{\left(x + 3 \right)}$
The result is: $\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 9 x - 27 \log{\left(x + 3 \right)}$
Add the constant of integration:

$\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 9 x - 27 \log{\left(x + 3 \right)}+ \mathrm{constant}$

The answer is:

$\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 9 x - 27 \log{\left(x + 3 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                              
 |                                               
 |    3                                    2    3
 |   x                                  3*x    x 
 | ----- dx = C - 27*log(3 + x) + 9*x - ---- + --
 | x + 3                                 2     3 
 |                                               
/

$${{2\,x^3-9\,x^2+54\,x}\over{6}}-27\,\log \left(x+3\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

47/6 - 27*log(4) + 27*log(3)

$$27\,\log 3-{{162\,\log 4-47}\over{6}}$$

47/6 - 27*log(4) + 27*log(3)

$$- 27 \log{\left(4 \right)} + \frac{47}{6} + 27 \log{\left(3 \right)}$$

Упростить

Numerical answer [src]

0.0659173771352483

0.0659173771352483

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+3) dx

Limits of integration:

The graph:

Piecewise:

The solution