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Solving integrals/ 1/(2x^2+3x+1)

Integral of 1/(2x^2+3x+1) dx

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

$1 \cdot \frac{1}{2 x^{2} + 3 x + 1} = \frac{2}{2 x + 1} - \frac{1}{x + 1}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2}{2 x + 1}\, dx = 2 \int \frac{1}{2 x + 1}\, dx$
  1. Let $u = 2 x + 1$ .
    
    Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{1}{4 u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{2 u}\, du = \frac{\int \frac{1}{u}\, du}{2}$
      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:
    
    $\frac{\log{\left(2 x + 1 \right)}}{2}$
  So, the result is: $\log{\left(2 x + 1 \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{x + 1}\right)\, dx = - \int \frac{1}{x + 1}\, dx$
  1. Let $u = x + 1$ .
    
    Then let $du = dx$ and substitute $du$ :
    
    $\int \frac{1}{u}\, du$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    Now substitute $u$ back in:
    
    $\log{\left(x + 1 \right)}$
  So, the result is: $- \log{\left(x + 1 \right)}$
The result is: $- \log{\left(x + 1 \right)} + \log{\left(2 x + 1 \right)}$
Add the constant of integration:

$- \log{\left(x + 1 \right)} + \log{\left(2 x + 1 \right)}+ \mathrm{constant}$

The answer is:

$- \log{\left(x + 1 \right)} + \log{\left(2 x + 1 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$\log \left(2\,x+1\right)-\log \left(x+1\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

log(3/2)

$$\log 3-\log 2$$

log(3/2)

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

Simplify

Numerical answer [src]

0.405465108108164

0.405465108108164

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/(2x^2+3x+1) dx

Limits of integration:

The graph:

Piecewise:

The solution