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Identical expressions
(arctg^ four (x)/(x^ two + one))
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(arctg to the power of four (x) divide by (x to the power of two plus one))
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(arctg to the power of 4(x)/(x to the power of 2+1))
arctg^4x/x^2+1
(arctg^4(x) divide by (x^2+1))
(arctg^4(x)/(x^2+1))dx
Similar expressions
(arctg^4(x)/(x^2-1))

Integral of (arctg^4(x)/(x^2+1)) dx

The solution

You have entered [src]

 oo            
  /            
 |             
 |      4      
 |  atan (x)   
 |  -------- dx
 |    2        
 |   x  + 1    
 |             
/              
-oo

$$\int\limits_{-\infty}^{\infty} \frac{\operatorname{atan}^{4}{\left(x \right)}}{x^{2} + 1}\, dx$$

Integral(atan(x)^4/(x^2 + 1), (x, -oo, oo))

Detail solution

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

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

$\int u^{4}\, du$
1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int u^{4}\, du = \frac{u^{5}}{5}$
Now substitute $u$ back in:

$\frac{\operatorname{atan}^{5}{\left(x \right)}}{5}$
Add the constant of integration:

$\frac{\operatorname{atan}^{5}{\left(x \right)}}{5}+ \mathrm{constant}$

The answer is:

$\frac{\operatorname{atan}^{5}{\left(x \right)}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                          
 |                           
 |     4                 5   
 | atan (x)          atan (x)
 | -------- dx = C + --------
 |   2                  5    
 |  x  + 1                   
 |                           
/

$$\int \frac{\operatorname{atan}^{4}{\left(x \right)}}{x^{2} + 1}\, dx = C + \frac{\operatorname{atan}^{5}{\left(x \right)}}{5}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  5
pi 
---
 80

$$\frac{\pi^{5}}{80}$$

  5
pi 
---
 80

$$\frac{\pi^{5}}{80}$$

pi^5/80

Numerical answer [src]

3.82524605981602

3.82524605981602

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

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Integral of (arctg^4(x)/(x^2+1)) dx

Limits of integration:

The graph:

Piecewise:

The solution