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Identical expressions
5x/(x+ two)(x^ two + one)
5x divide by (x plus 2)(x squared plus 1)
5x divide by (x plus two)(x to the power of two plus one)
5x/(x+2)(x2+1)
5x/x+2x2+1
5x/(x+2)(x²+1)
5x/(x+2)(x to the power of 2+1)
5x/x+2x^2+1
5x divide by (x+2)(x^2+1)
5x/(x+2)(x^2+1)dx
Similar expressions
5x/(x-2)(x^2+1)
5x/(x+2)(x^2-1)

Solving integrals/ 5x/(x+2)(x^2+1)

Integral of 5x/(x+2)(x^2+1) dx

The solution

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  1                
  /                
 |                 
 |      / 2    \   
 |  5*x*\x  + 1/   
 |  ------------ dx
 |     x + 2       
 |                 
/                  
0

$$\int\limits_{0}^{1} \frac{5 x \left(x^{2} + 1\right)}{x + 2}\, dx$$

Integral(5*x*(x^2 + 1)/(x + 2), (x, 0, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{5 x \left(x^{2} + 1\right)}{x + 2}\, dx = 5 \int \frac{x \left(x^{2} + 1\right)}{x + 2}\, dx$
1. There are multiple ways to do this integral.
  Method #1
  1. Rewrite the integrand:
    
    $\frac{x \left(x^{2} + 1\right)}{x + 2} = x^{2} - 2 x + 5 - \frac{10}{x + 2}$
  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:
      
      $\int \left(- 2 x\right)\, dx = - 2 \int x\, dx$
      
      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: $- x^{2}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 5\, dx = 5 x$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{10}{x + 2}\right)\, dx = - 10 \int \frac{1}{x + 2}\, dx$
      
      Let $u = x + 2$ .
      
      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:
      
      $\log{\left(x + 2 \right)}$
      
      So, the result is: $- 10 \log{\left(x + 2 \right)}$
    The result is: $\frac{x^{3}}{3} - x^{2} + 5 x - 10 \log{\left(x + 2 \right)}$
  Method #2
  1. Rewrite the integrand:
    
    $\frac{x \left(x^{2} + 1\right)}{x + 2} = \frac{x^{3} + x}{x + 2}$
  2. Rewrite the integrand:
    
    $\frac{x^{3} + x}{x + 2} = x^{2} - 2 x + 5 - \frac{10}{x + 2}$
  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:
      
      $\int \left(- 2 x\right)\, dx = - 2 \int x\, dx$
      
      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: $- x^{2}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 5\, dx = 5 x$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{10}{x + 2}\right)\, dx = - 10 \int \frac{1}{x + 2}\, dx$
      
      Let $u = x + 2$ .
      
      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:
      
      $\log{\left(x + 2 \right)}$
      
      So, the result is: $- 10 \log{\left(x + 2 \right)}$
    The result is: $\frac{x^{3}}{3} - x^{2} + 5 x - 10 \log{\left(x + 2 \right)}$
  Method #3
  1. Rewrite the integrand:
    
    $\frac{x \left(x^{2} + 1\right)}{x + 2} = \frac{x^{3}}{x + 2} + \frac{x}{x + 2}$
  2. Integrate term-by-term:
    1. Rewrite the integrand:
      
      $\frac{x^{3}}{x + 2} = x^{2} - 2 x + 4 - \frac{8}{x + 2}$
    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:
      
      $\int \left(- 2 x\right)\, dx = - 2 \int x\, dx$
      
      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: $- x^{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 4\, dx = 4 x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{8}{x + 2}\right)\, dx = - 8 \int \frac{1}{x + 2}\, dx$
      
      Let $u = x + 2$ .
      
      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:
      
      $\log{\left(x + 2 \right)}$
      
      So, the result is: $- 8 \log{\left(x + 2 \right)}$
      
      The result is: $\frac{x^{3}}{3} - x^{2} + 4 x - 8 \log{\left(x + 2 \right)}$
    1. Rewrite the integrand:
      
      $\frac{x}{x + 2} = 1 - \frac{2}{x + 2}$
    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:
      
      $\int \left(- \frac{2}{x + 2}\right)\, dx = - 2 \int \frac{1}{x + 2}\, dx$
      
      Let $u = x + 2$ .
      
      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:
      
      $\log{\left(x + 2 \right)}$
      
      So, the result is: $- 2 \log{\left(x + 2 \right)}$
      
      The result is: $x - 2 \log{\left(x + 2 \right)}$
    The result is: $\frac{x^{3}}{3} - x^{2} + 5 x - 10 \log{\left(x + 2 \right)}$
So, the result is: $\frac{5 x^{3}}{3} - 5 x^{2} + 25 x - 50 \log{\left(x + 2 \right)}$
Add the constant of integration:

$\frac{5 x^{3}}{3} - 5 x^{2} + 25 x - 50 \log{\left(x + 2 \right)}+ \mathrm{constant}$

The answer is:

$\frac{5 x^{3}}{3} - 5 x^{2} + 25 x - 50 \log{\left(x + 2 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                        
 |                                                         
 |     / 2    \                                           3
 | 5*x*\x  + 1/                             2          5*x 
 | ------------ dx = C - 50*log(2 + x) - 5*x  + 25*x + ----
 |    x + 2                                             3  
 |                                                         
/

$$5\,\left({{x^3-3\,x^2+15\,x}\over{3}}-10\,\log \left(x+2\right) \right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

65/3 - 50*log(3) + 50*log(2)

$$5\,\left(10\,\log 2-{{30\,\log 3-13}\over{3}}\right)$$

65/3 - 50*log(3) + 50*log(2)

$$- 50 \log{\left(3 \right)} + \frac{65}{3} + 50 \log{\left(2 \right)}$$

Simplify

Numerical answer [src]

1.39341126125845

1.39341126125845

The graph

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 5x/(x+2)(x^2+1) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3