Integral of x(arctgx/2) dx

The solution

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  1             
  /             
 |              
 |  x*acot(x)   
 |  --------- dx
 |      2       
 |              
/               
0

$$\int\limits_{0}^{1} \frac{x \operatorname{acot}{\left(x \right)}}{2}\, dx$$

Integral(x*acot(x)/2, (x, 0, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{x \operatorname{acot}{\left(x \right)}}{2}\, dx = \frac{\int x \operatorname{acot}{\left(x \right)}\, dx}{2}$
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \operatorname{acot}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = x$ .
  
  Then $\operatorname{du}{\left(x \right)} = - \frac{1}{x^{2} + 1}$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{x^{2}}{2 \left(x^{2} + 1\right)}\right)\, dx = - \frac{\int \frac{x^{2}}{x^{2} + 1}\, dx}{2}$
  1. Rewrite the integrand:
    
    $\frac{x^{2}}{x^{2} + 1} = 1 - \frac{1}{x^{2} + 1}$
  2. Integrate term-by-term:
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x^{2} + 1}\right)\, dx = - \int \frac{1}{x^{2} + 1}\, dx$
      1. The integral of $\frac{1}{x^{2} + 1}$ is $\operatorname{atan}{\left(x \right)}$ .
      So, the result is: $- \operatorname{atan}{\left(x \right)}$
    The result is: $x - \operatorname{atan}{\left(x \right)}$
  So, the result is: $- \frac{x}{2} + \frac{\operatorname{atan}{\left(x \right)}}{2}$
So, the result is: $\frac{x^{2} \operatorname{acot}{\left(x \right)}}{4} + \frac{x}{4} - \frac{\operatorname{atan}{\left(x \right)}}{4}$
Add the constant of integration:

$\frac{x^{2} \operatorname{acot}{\left(x \right)}}{4} + \frac{x}{4} - \frac{\operatorname{atan}{\left(x \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \operatorname{acot}{\left(x \right)}}{4} + \frac{x}{4} - \frac{\operatorname{atan}{\left(x \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                           
 |                                   2        
 | x*acot(x)          atan(x)   x   x *acot(x)
 | --------- dx = C - ------- + - + ----------
 |     2                 4      4       4     
 |                                            
/

$${{{{x-\arctan x}\over{2}}+{{x^2\,{\rm arccot}\; x}\over{2}}}\over{2 }}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/4

$${{1}\over{4}}$$

1/4

$$\frac{1}{4}$$

Упростить

Numerical answer [src]

0.25

0.25

The graph

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

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Identical expressions

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Integral of x(arctgx/2) dx

Limits of integration:

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The solution