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

The solution

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

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

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

Detail solution

Rewrite the integrand:

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

$\frac{x}{2} - \frac{\log{\left(2 x + 1 \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{x}{2} - \frac{\log{\left(2 x + 1 \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                                  
 |    x             x   log(1 + 2*x)
 | ------- dx = C + - - ------------
 | 2*x + 1          2        4      
 |                                  
/

$$\int \frac{x}{2 x + 1}\, dx = C + \frac{x}{2} - \frac{\log{\left(2 x + 1 \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   log(3)
- - ------
2     4

$$\frac{1}{2} - \frac{\log{\left(3 \right)}}{4}$$

1   log(3)
- - ------
2     4

$$\frac{1}{2} - \frac{\log{\left(3 \right)}}{4}$$

1/2 - log(3)/4

Numerical answer [src]

0.225346927832973

0.225346927832973

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/(2*x+1) dx

Limits of integration:

The graph:

Piecewise:

The solution