Integral of dx/(2x+1) dx

The solution

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

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

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

Detail solution

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:

$\frac{\log{\left(2 x + 1 \right)}}{2}$
Now simplify:

$\frac{\log{\left(2 x + 1 \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

log(5)
------
  2

$$\frac{\log{\left(5 \right)}}{2}$$

log(5)
------
  2

$$\frac{\log{\left(5 \right)}}{2}$$

log(5)/2

Numerical answer [src]

0.80471895621705

0.80471895621705

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

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Integral of dx/(2x+1) dx

Limits of integration:

The graph:

Piecewise:

The solution