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2dx/(2x-3)2

Integral of 2dx/(2x+3)2 dx

The solution

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

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

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

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int 2 \frac{2}{2 x + 3}\, dx = 2 \int \frac{2}{2 x + 3}\, dx$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2}{2 x + 3}\, dx = 2 \int \frac{1}{2 x + 3}\, dx$
  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: $\log{\left(2 x + 3 \right)}$
So, the result is: $2 \log{\left(2 x + 3 \right)}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-2*log(3) + 2*log(5)

$$- 2 \log{\left(3 \right)} + 2 \log{\left(5 \right)}$$

-2*log(3) + 2*log(5)

$$- 2 \log{\left(3 \right)} + 2 \log{\left(5 \right)}$$

-2*log(3) + 2*log(5)

Numerical answer [src]

1.02165124753198

1.02165124753198

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

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Integral of 2dx/(2x+3)2 dx

Limits of integration:

The graph:

Piecewise:

The solution