Integral of x/(2+x) dx

The solution

You have entered [src]

  1         
  /         
 |          
 |    x     
 |  ----- dx
 |  2 + x   
 |          
/           
0

$$\int\limits_{0}^{1} \frac{x}{x + 2}\, dx$$

Integral(x/(2 + x), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{x}{x + 2} = 1 - \frac{2}{x + 2}$
Integrate term-by-term:
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:
  
  $\int \left(- \frac{2}{x + 2}\right)\, dx = - 2 \int \frac{1}{x + 2}\, dx$
  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: $- 2 \log{\left(x + 2 \right)}$
The result is: $x - 2 \log{\left(x + 2 \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

1 - 2*log(3) + 2*log(2)

$$- 2 \log{\left(3 \right)} + 1 + 2 \log{\left(2 \right)}$$

1 - 2*log(3) + 2*log(2)

$$- 2 \log{\left(3 \right)} + 1 + 2 \log{\left(2 \right)}$$

1 - 2*log(3) + 2*log(2)

Numerical answer [src]

0.189069783783671

0.189069783783671

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/(2+x) dx

Limits of integration:

The graph:

Piecewise:

The solution