Integral of x/(2x-3) dx

The solution

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  1           
  /           
 |            
 |     x      
 |  ------- dx
 |  2*x - 3   
 |            
/             
0

$$\int\limits_{0}^{1} \frac{x}{2 x - 3}\, dx$$

Integral(x/(2*x - 3), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{x}{2 x - 3} = \frac{1}{2} + \frac{3}{2 \left(2 x - 3\right)}$
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:
  
  $\int \frac{3}{2 \left(2 x - 3\right)}\, dx = \frac{3 \int \frac{1}{2 x - 3}\, dx}{2}$
  1. Let $u = 2 x - 3$ .
    
    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:
    
    $\frac{\log{\left(2 x - 3 \right)}}{2}$
  So, the result is: $\frac{3 \log{\left(2 x - 3 \right)}}{4}$
The result is: $\frac{x}{2} + \frac{3 \log{\left(2 x - 3 \right)}}{4}$
Add the constant of integration:

$\frac{x}{2} + \frac{3 \log{\left(2 x - 3 \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{x}{2} + \frac{3 \log{\left(2 x - 3 \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                                     
 |    x             x   3*log(-3 + 2*x)
 | ------- dx = C + - + ---------------
 | 2*x - 3          2          4       
 |                                     
/

$$\int \frac{x}{2 x - 3}\, dx = C + \frac{x}{2} + \frac{3 \log{\left(2 x - 3 \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

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

Numerical answer [src]

-0.323959216501082

-0.323959216501082

The graph

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

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Integral of x/(2x-3) dx

Limits of integration:

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Piecewise:

The solution