Integral of 1/(3x+2) dx

The solution

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   ___            
 \/ 3             
   /              
  |               
  |        1      
  |   1*------- dx
  |     3*x + 2   
  |               
 /                
  ___             
\/ 3              
-----             
  3

$$\int\limits_{\frac{\sqrt{3}}{3}}^{\sqrt{3}} 1 \cdot \frac{1}{3 x + 2}\, dx$$

Integral(1/(3*x + 2), (x, sqrt(3)/3, sqrt(3)))

Detail solution

Let $u = 3 x + 2$ .

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

$\int \frac{1}{9 u}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{3 u}\, du = \frac{\int \frac{1}{u}\, du}{3}$
  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:

$\frac{\log{\left(3 x + 2 \right)}}{3}$
Now simplify:

$\frac{\log{\left(3 x + 2 \right)}}{3}$
Add the constant of integration:

$\frac{\log{\left(3 x + 2 \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(3 x + 2 \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                                
 |      1             log(3*x + 2)
 | 1*------- dx = C + ------------
 |   3*x + 2               3      
 |                                
/

$$\int 1 \cdot \frac{1}{3 x + 2}\, dx = C + \frac{\log{\left(3 x + 2 \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

     /      ___\      /        ___\
  log\2 + \/ 3 /   log\2 + 3*\/ 3 /
- -------------- + ----------------
        3                 3

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

     /      ___\      /        ___\
  log\2 + \/ 3 /   log\2 + 3*\/ 3 /
- -------------- + ----------------
        3                 3

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

Упростить

Numerical answer [src]

0.218862866842353

0.218862866842353

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 1/(3x+2) dx

Limits of integration:

The graph:

Piecewise:

The solution