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

The solution

You have entered [src]

 oo            
  /            
 |             
 |      2      
 |  atan (x)   
 |  -------- dx
 |        2    
 |   1 + x     
 |             
/              
1

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

Integral(atan(x)^2/(1 + x^2), (x, 1, oo))

Detail solution

Let $u = \operatorname{atan}{\left(x \right)}$ .

Then let $du = \frac{dx}{x^{2} + 1}$ and substitute $du$ :

$\int u^{2}\, du$
1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int u^{2}\, du = \frac{u^{3}}{3}$
Now substitute $u$ back in:

$\frac{\operatorname{atan}^{3}{\left(x \right)}}{3}$
Add the constant of integration:

$\frac{\operatorname{atan}^{3}{\left(x \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{\operatorname{atan}^{3}{\left(x \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$\int \frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{2} + 1}\, dx = C + \frac{\operatorname{atan}^{3}{\left(x \right)}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    3
7*pi 
-----
 192

$$\frac{7 \pi^{3}}{192}$$

    3
7*pi 
-----
 192

$$\frac{7 \pi^{3}}{192}$$

7*pi^3/192

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

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

Limits of integration:

The graph:

Piecewise:

The solution