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Integral of d{x}:
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Integral of y^2
Integral of xe^(x^2)
Derivative of:
arctgx/(1+x^2)
Identical expressions
arctgx/(one +x^ two)
arctgx divide by (1 plus x squared )
arctgx divide by (one plus x to the power of two)
arctgx/(1+x2)
arctgx/1+x2
arctgx/(1+x²)
arctgx/(1+x to the power of 2)
arctgx/1+x^2
arctgx divide by (1+x^2)
arctgx/(1+x^2)dx
Similar expressions
arctgx/(1-x^2)

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

The solution

You have entered [src]

  1           
  /           
 |            
 |  acot(x)   
 |  ------- dx
 |        2   
 |   1 + x    
 |            
/             
0

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \operatorname{acot}{\left(x \right)}$ .
  
  Then let $du = - \frac{dx}{x^{2} + 1}$ and substitute $- du$ :
  
  $\int u\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- u\right)\, du = - \int u\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
    So, the result is: $- \frac{u^{2}}{2}$
  Now substitute $u$ back in:
  
  $- \frac{\operatorname{acot}^{2}{\left(x \right)}}{2}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \operatorname{acot}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2} + 1}$ .
  
  Then $\operatorname{du}{\left(x \right)} = - \frac{1}{x^{2} + 1}$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
  Now evaluate the sub-integral.
2. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = - \operatorname{atan}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2} + 1}$ .
  
  Then $\operatorname{du}{\left(x \right)} = - \frac{1}{x^{2} + 1}$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
  Now evaluate the sub-integral.
3. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = - \operatorname{atan}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2} + 1}$ .
  
  Then $\operatorname{du}{\left(x \right)} = - \frac{1}{x^{2} + 1}$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
  Now evaluate the sub-integral.
4. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = - \operatorname{atan}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2} + 1}$ .
  
  Then $\operatorname{du}{\left(x \right)} = - \frac{1}{x^{2} + 1}$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
  Now evaluate the sub-integral.
5. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = - \operatorname{atan}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2} + 1}$ .
  
  Then $\operatorname{du}{\left(x \right)} = - \frac{1}{x^{2} + 1}$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
  Now evaluate the sub-integral.
6. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\operatorname{atan}{\left(x \right)}}{x^{2} + 1}\right)\, dx = - \int \frac{\operatorname{atan}{\left(x \right)}}{x^{2} + 1}\, dx$
  1. Let $u = \operatorname{atan}{\left(x \right)}$ .
    
    Then let $du = \frac{dx}{x^{2} + 1}$ and substitute $du$ :
    
    $\int u\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
    Now substitute $u$ back in:
    
    $\frac{\operatorname{atan}^{2}{\left(x \right)}}{2}$
  So, the result is: $- \frac{\operatorname{atan}^{2}{\left(x \right)}}{2}$
Add the constant of integration:

$- \frac{\operatorname{acot}^{2}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{\operatorname{acot}^{2}{\left(x \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                         
 |                      2   
 | acot(x)          acot (x)
 | ------- dx = C - --------
 |       2             2    
 |  1 + x                   
 |                          
/

$$-{{\left({\rm arccot}\; x\right)^2}\over{2}}$$

Simplify

The answer [src]

    2
3*pi 
-----
  32

$${{3\,\pi^2}\over{32}}$$

    2
3*pi 
-----
  32

$$\frac{3 \pi^{2}}{32}$$

Упростить

Numerical answer [src]

0.925275412602127

0.925275412602127

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2