Integral of 2*x/(x+1) dx

The solution

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

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

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

Detail solution

Rewrite the integrand:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

2 - 2*log(2)

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

2 - 2*log(2)

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

2 - 2*log(2)

Numerical answer [src]

0.613705638880109

0.613705638880109

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

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Integral of 2*x/(x+1) dx

Limits of integration:

The graph:

Piecewise:

The solution