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arctg^3x/(1-x^2)

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

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

The solution

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 oo            
  /            
 |             
 |      3      
 |  atan (x)   
 |  -------- dx
 |        2    
 |   1 + x     
 |             
/              
0

$$\int\limits_{0}^{\infty} \frac{\operatorname{atan}^{3}{\left(x \right)}}{x^{2} + 1}\, dx$$

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

Detail solution

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

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

$\int u^{3}\, du$
1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int u^{3}\, du = \frac{u^{4}}{4}$
Now substitute $u$ back in:

$\frac{\operatorname{atan}^{4}{\left(x \right)}}{4}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                          
 |                           
 |     3                 4   
 | atan (x)          atan (x)
 | -------- dx = C + --------
 |       2              4    
 |  1 + x                    
 |                           
/

$$\int \frac{\operatorname{atan}^{3}{\left(x \right)}}{x^{2} + 1}\, dx = C + \frac{\operatorname{atan}^{4}{\left(x \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  4
pi 
---
 64

$$\frac{\pi^{4}}{64}$$

  4
pi 
---
 64

$$\frac{\pi^{4}}{64}$$

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

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

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The solution