Integral of x/(3x-1) dx

The solution

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

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

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

Detail solution

Rewrite the integrand:

$\frac{x}{3 x - 1} = \frac{1}{3} + \frac{1}{3 \left(3 x - 1\right)}$
Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \frac{1}{3}\, dx = \frac{x}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{3 \left(3 x - 1\right)}\, dx = \frac{\int \frac{1}{3 x - 1}\, dx}{3}$
  1. 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:
    
    $\frac{\log{\left(3 x - 1 \right)}}{3}$
  So, the result is: $\frac{\log{\left(3 x - 1 \right)}}{9}$
The result is: $\frac{x}{3} + \frac{\log{\left(3 x - 1 \right)}}{9}$
Add the constant of integration:

$\frac{x}{3} + \frac{\log{\left(3 x - 1 \right)}}{9}+ \mathrm{constant}$

The answer is:

$\frac{x}{3} + \frac{\log{\left(3 x - 1 \right)}}{9}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                  
 |                                   
 |    x             x   log(-1 + 3*x)
 | ------- dx = C + - + -------------
 | 3*x - 1          3         9      
 |                                   
/

$$\int \frac{x}{3 x - 1}\, dx = C + \frac{x}{3} + \frac{\log{\left(3 x - 1 \right)}}{9}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$\text{NaN}$$

nan

$$\text{NaN}$$

nan

Numerical answer [src]

13.3999878764332

13.3999878764332

The graph

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

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Integral of x/(3x-1) dx

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The solution