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

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

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The solution

You have entered [src] 
 oo            
  /            
 |             
 |      3      
 |  atan (x)   
 |  -------- dx
 |        2    
 |   1 + x     
 |             
/              
0
∫0∞atan⁡3(x)x2+1 dx\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

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

    Then let du=dxx2+1du = \frac{dx}{x^{2} + 1} and substitute dudu:

    ∫u3 du\int u^{3}\, du

    1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫u3 du=u44\int u^{3}\, du = \frac{u^{4}}{4}

    Now substitute uu back in:

    atan⁡4(x)4\frac{\operatorname{atan}^{4}{\left(x \right)}}{4}

  2. Add the constant of integration:

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

The answer is:

atan⁡4(x)4+constant\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                    
 |                           
/
∫atan⁡3(x)x2+1 dx=C+atan⁡4(x)4\int \frac{\operatorname{atan}^{3}{\left(x \right)}}{x^{2} + 1}\, dx = C + \frac{\operatorname{atan}^{4}{\left(x \right)}}{4}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
  4
pi 
---
 64
π464\frac{\pi^{4}}{64} 
=
= 
  4
pi 
---
 64
π464\frac{\pi^{4}}{64}
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The graph
Integral of arctg^3x/(1+x^2) dx

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

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