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

The solution

You have entered [src]

  1                
  /                
 |                 
 |  log(atan(x))   
 |  ------------ dx
 |          2      
 |     1 + x       
 |                 
/                  
0

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

Integral(log(atan(x))/(1 + x^2), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \log{\left(\operatorname{atan}{\left(x \right)} \right)}$ .
  
  Then let $du = \frac{dx}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}$ and substitute $du$ :
  
  $\int u e^{u}\, du$
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
    
    Then $\operatorname{du}{\left(u \right)} = 1$ .
    
    To find $v{\left(u \right)}$ :
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    Now evaluate the sub-integral.
  2. The integral of the exponential function is itself.
    
    $\int e^{u}\, du = e^{u}$
  Now substitute $u$ back in:
  
  $\log{\left(\operatorname{atan}{\left(x \right)} \right)} \operatorname{atan}{\left(x \right)} - \operatorname{atan}{\left(x \right)}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \log{\left(\operatorname{atan}{\left(x \right)} \right)}$ and let $\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2} + 1}$ .
  
  Then $\operatorname{du}{\left(x \right)} = \frac{1}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}$ .
  
  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. The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
Now simplify:

$\left(\log{\left(\operatorname{atan}{\left(x \right)} \right)} - 1\right) \operatorname{atan}{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\arctan x\,\log \arctan x-\arctan x$$

Simplify

The answer [src]

             /pi\
       pi*log|--|
  pi         \4 /
- -- + ----------
  4        4

$${{\pi\,\log \left({{\pi}\over{4}}\right)}\over{4}}-{{\pi}\over{4}}$$

             /pi\
       pi*log|--|
  pi         \4 /
- -- + ----------
  4        4

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

Simplify

Numerical answer [src]

-0.97512245861696

-0.97512245861696

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2