Integral of 1/(3x+1) dx

The solution

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  1           
  /           
 |            
 |     1      
 |  ------- dx
 |  3*x + 1   
 |            
/             
0

$$\int\limits_{0}^{1} \frac{1}{3 x + 1}\, dx$$

Integral(1/(3*x + 1), (x, 0, 1))

Detail solution

Let $u = 3 x + 1$ .

Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :

$\int \frac{1}{3 u}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{3}$
  1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
  So, the result is: $\frac{\log{\left(u \right)}}{3}$
Now substitute $u$ back in:

$\frac{\log{\left(3 x + 1 \right)}}{3}$
Now simplify:

$\frac{\log{\left(3 x + 1 \right)}}{3}$
Add the constant of integration:

$\frac{\log{\left(3 x + 1 \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(3 x + 1 \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                             
 |                              
 |    1             log(3*x + 1)
 | ------- dx = C + ------------
 | 3*x + 1               3      
 |                              
/

$$\int \frac{1}{3 x + 1}\, dx = C + \frac{\log{\left(3 x + 1 \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

log(4)
------
  3

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

log(4)
------
  3

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

log(4)/3

Numerical answer [src]

0.462098120373297

0.462098120373297

The graph

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

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Identical expressions

Similar expressions

Integral of 1/(3x+1) dx

Limits of integration:

The graph:

Piecewise:

The solution