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Derivative of:
2*x/(x+2)
Identical expressions
two *x/(x+ two)
2 multiply by x divide by (x plus 2)
two multiply by x divide by (x plus two)
2x/(x+2)
2x/x+2
2*x divide by (x+2)
2*x/(x+2)dx
Similar expressions
(x^4+3x^3+2x)/((x+2)(x+4))
2*x/(x-2)

Integral of 2*x/(x+2) dx

The solution

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

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

Integral((2*x)/(x + 2), (x, 0, 2))

Detail solution

Rewrite the integrand:

$\frac{2 x}{x + 2} = 2 - \frac{4}{x + 2}$
Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 2\, dx = 2 x$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{4}{x + 2}\right)\, dx = - 4 \int \frac{1}{x + 2}\, dx$
  1. Let $u = x + 2$ .
    
    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:
    
    $\log{\left(x + 2 \right)}$
  So, the result is: $- 4 \log{\left(x + 2 \right)}$
The result is: $2 x - 4 \log{\left(x + 2 \right)}$
Add the constant of integration:

$2 x - 4 \log{\left(x + 2 \right)}+ \mathrm{constant}$

The answer is:

$2 x - 4 \log{\left(x + 2 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                                  
 |  2*x                             
 | ----- dx = C - 4*log(2 + x) + 2*x
 | x + 2                            
 |                                  
/

$$\int \frac{2 x}{x + 2}\, dx = C + 2 x - 4 \log{\left(x + 2 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

4 - 4*log(4) + 4*log(2)

$$- 4 \log{\left(4 \right)} + 4 \log{\left(2 \right)} + 4$$

4 - 4*log(4) + 4*log(2)

$$- 4 \log{\left(4 \right)} + 4 \log{\left(2 \right)} + 4$$

4 - 4*log(4) + 4*log(2)

Numerical answer [src]

1.22741127776022

1.22741127776022

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

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Identical expressions

Similar expressions

Integral of 2*x/(x+2) dx

Limits of integration:

The graph:

Piecewise:

The solution