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

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

The solution

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  1                      
  /                      
 |                       
 |           1           
 |  1*---------------- dx
 |            / 2    \   
 |    (x + 1)*\x  + 1/   
 |                       
/                        
0

$$\int\limits_{0}^{1} 1 \cdot \frac{1}{\left(x + 1\right) \left(x^{2} + 1\right)}\, dx$$

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $1 \cdot \frac{1}{\left(x + 1\right) \left(x^{2} + 1\right)} = - \frac{x - 1}{2 \left(x^{2} + 1\right)} + \frac{1}{2 \left(x + 1\right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{x - 1}{2 \left(x^{2} + 1\right)}\right)\, dx = - \frac{\int \frac{x - 1}{x^{2} + 1}\, dx}{2}$
    1. Rewrite the integrand:
      
      $\frac{x - 1}{x^{2} + 1} = \frac{x}{x^{2} + 1} - \frac{1}{x^{2} + 1}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{x}{x^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + 1}\, dx}{2}$
      
      Let $u = x^{2} + 1$ .
      
      Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{2 u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x^{2} + 1 \right)}$
      
      So, the result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x^{2} + 1}\right)\, dx = - \int \frac{1}{x^{2} + 1}\, dx$
      
      The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
      
      So, the result is: $- \operatorname{atan}{\left(x \right)}$
      
      The result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2} - \operatorname{atan}{\left(x \right)}$
    So, the result is: $- \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\operatorname{atan}{\left(x \right)}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{2 \left(x + 1\right)}\, dx = \frac{\int \frac{1}{x + 1}\, dx}{2}$
    1. Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      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: $\frac{\log{\left(x + 1 \right)}}{2}$
  The result is: $\frac{\log{\left(x + 1 \right)}}{2} - \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\operatorname{atan}{\left(x \right)}}{2}$
Method #2
1. Rewrite the integrand:
  
  $1 \cdot \frac{1}{\left(x + 1\right) \left(x^{2} + 1\right)} = \frac{1}{x^{3} + x^{2} + x + 1}$
2. Rewrite the integrand:
  
  $\frac{1}{x^{3} + x^{2} + x + 1} = - \frac{x - 1}{2 \left(x^{2} + 1\right)} + \frac{1}{2 \left(x + 1\right)}$
3. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{x - 1}{2 \left(x^{2} + 1\right)}\right)\, dx = - \frac{\int \frac{x - 1}{x^{2} + 1}\, dx}{2}$
    1. Rewrite the integrand:
      
      $\frac{x - 1}{x^{2} + 1} = \frac{x}{x^{2} + 1} - \frac{1}{x^{2} + 1}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{x}{x^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + 1}\, dx}{2}$
      
      Let $u = x^{2} + 1$ .
      
      Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{2 u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x^{2} + 1 \right)}$
      
      So, the result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x^{2} + 1}\right)\, dx = - \int \frac{1}{x^{2} + 1}\, dx$
      
      The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
      
      So, the result is: $- \operatorname{atan}{\left(x \right)}$
      
      The result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2} - \operatorname{atan}{\left(x \right)}$
    So, the result is: $- \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\operatorname{atan}{\left(x \right)}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{2 \left(x + 1\right)}\, dx = \frac{\int \frac{1}{x + 1}\, dx}{2}$
    1. Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      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: $\frac{\log{\left(x + 1 \right)}}{2}$
  The result is: $\frac{\log{\left(x + 1 \right)}}{2} - \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\operatorname{atan}{\left(x \right)}}{2}$
Method #3
1. Rewrite the integrand:
  
  $1 \cdot \frac{1}{\left(x + 1\right) \left(x^{2} + 1\right)} = \frac{1}{x^{3} + x^{2} + x + 1}$
2. Rewrite the integrand:
  
  $\frac{1}{x^{3} + x^{2} + x + 1} = - \frac{x - 1}{2 \left(x^{2} + 1\right)} + \frac{1}{2 \left(x + 1\right)}$
3. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{x - 1}{2 \left(x^{2} + 1\right)}\right)\, dx = - \frac{\int \frac{x - 1}{x^{2} + 1}\, dx}{2}$
    1. Rewrite the integrand:
      
      $\frac{x - 1}{x^{2} + 1} = \frac{x}{x^{2} + 1} - \frac{1}{x^{2} + 1}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{x}{x^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + 1}\, dx}{2}$
      
      Let $u = x^{2} + 1$ .
      
      Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{2 u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x^{2} + 1 \right)}$
      
      So, the result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x^{2} + 1}\right)\, dx = - \int \frac{1}{x^{2} + 1}\, dx$
      
      The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
      
      So, the result is: $- \operatorname{atan}{\left(x \right)}$
      
      The result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2} - \operatorname{atan}{\left(x \right)}$
    So, the result is: $- \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\operatorname{atan}{\left(x \right)}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{2 \left(x + 1\right)}\, dx = \frac{\int \frac{1}{x + 1}\, dx}{2}$
    1. Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      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: $\frac{\log{\left(x + 1 \right)}}{2}$
  The result is: $\frac{\log{\left(x + 1 \right)}}{2} - \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\operatorname{atan}{\left(x \right)}}{2}$
Add the constant of integration:

$\frac{\log{\left(x + 1 \right)}}{2} - \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\operatorname{atan}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(x + 1 \right)}}{2} - \frac{\log{\left(x^{2} + 1 \right)}}{4} + \frac{\operatorname{atan}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                              
 |                                                       /     2\
 |          1                  atan(x)   log(1 + x)   log\1 + x /
 | 1*---------------- dx = C + ------- + ---------- - -----------
 |           / 2    \             2          2             4     
 |   (x + 1)*\x  + 1/                                            
 |                                                               
/

$$-{{\log \left(x^2+1\right)}\over{4}}+{{\log \left(x+1\right)}\over{ 2}}+{{\arctan x}\over{2}}$$

Simplify

The graph

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The answer [src]

log(2)   pi
------ + --
  4      8

$${{\log 2}\over{4}}+{{\pi}\over{8}}$$

log(2)   pi
------ + --
  4      8

$$\frac{\log{\left(2 \right)}}{4} + \frac{\pi}{8}$$

Упростить

Numerical answer [src]

0.56598587683871

0.56598587683871

The graph

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

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How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3