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Identical expressions
(x-arctg^ four (x))/(one +x^ two)
(x minus arctg to the power of 4(x)) divide by (1 plus x squared )
(x minus arctg to the power of four (x)) divide by (one plus x to the power of two)
(x-arctg4(x))/(1+x2)
x-arctg4x/1+x2
(x-arctg⁴(x))/(1+x²)
(x-arctg to the power of 4(x))/(1+x to the power of 2)
x-arctg^4x/1+x^2
(x-arctg^4(x)) divide by (1+x^2)
(x-arctg^4(x))/(1+x^2)dx
Similar expressions
(x-arctg^4(x))/(1-x^2)
(x+arctg^4(x))/(1+x^2)

Solving integrals/ (x-arctg^4(x))/(1+x^2)

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

The solution

You have entered [src]

   ___               
 \/ 3                
   /                 
  |                  
  |           4      
  |   x - atan (x)   
  |   ------------ dx
  |           2      
  |      1 + x       
  |                  
 /                   
 0

$$\int\limits_{0}^{\sqrt{3}} \frac{x - \operatorname{atan}^{4}{\left(x \right)}}{x^{2} + 1}\, dx$$

Integral((x - atan(x)^4)/(1 + x^2), (x, 0, sqrt(3)))

Detail solution

Rewrite the integrand:

$\frac{x - \operatorname{atan}^{4}{\left(x \right)}}{x^{2} + 1} = \frac{x}{x^{2} + 1} - \frac{\operatorname{atan}^{4}{\left(x \right)}}{x^{2} + 1}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x}{x^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + 1}\, dx}{2}$
  1. Let $u = x^{2} + 1$ .
    
    Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{1}{2 u}\, du$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    Now substitute $u$ back in:
    
    $\log{\left(x^{2} + 1 \right)}$
  So, the result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\operatorname{atan}^{4}{\left(x \right)}}{x^{2} + 1}\right)\, dx = - \int \frac{\operatorname{atan}^{4}{\left(x \right)}}{x^{2} + 1}\, dx$
  1. 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:
    
    $\frac{\operatorname{atan}^{5}{\left(x \right)}}{5}$
  So, the result is: $- \frac{\operatorname{atan}^{5}{\left(x \right)}}{5}$
The result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2} - \frac{\operatorname{atan}^{5}{\left(x \right)}}{5}$
Add the constant of integration:

$\frac{\log{\left(x^{2} + 1 \right)}}{2} - \frac{\operatorname{atan}^{5}{\left(x \right)}}{5}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(x^{2} + 1 \right)}}{2} - \frac{\operatorname{atan}^{5}{\left(x \right)}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                    5 
log(8)   log(2)   pi  
------ - ------ - ----
  2        2      1215

$$- \frac{\log{\left(2 \right)}}{2} - \frac{\pi^{5}}{1215} + \frac{\log{\left(8 \right)}}{2}$$

                    5 
log(8)   log(2)   pi  
------ - ------ - ----
  2        2      1215

$$- \frac{\log{\left(2 \right)}}{2} - \frac{\pi^{5}}{1215} + \frac{\log{\left(8 \right)}}{2}$$

Упростить

Numerical answer [src]

0.441279127238726

0.441279127238726

The graph

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution