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Similar expressions
arcctg(2x)/(pi^2(4x^2-1))

Solving integrals/ arcctg(2x)/(pi^2(4x^2+1))

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

The solution

You have entered [src]

 oo                  
  /                  
 |                   
 |    acot(2*x)      
 |  -------------- dx
 |    2 /   2    \   
 |  pi *\4*x  + 1/   
 |                   
/                    
1/2

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

Integral(acot(2*x)/((pi^2*(4*x^2 + 1))), (x, 1/2, oo))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \operatorname{acot}{\left(2 x \right)}$ .
  
  Then let $du = - \frac{2 dx}{4 x^{2} + 1}$ and substitute $- \frac{du}{2 \pi^{2}}$ :
  
  $\int \left(- \frac{u}{2 \pi^{2}}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int u\, du = - \frac{\int u\, du}{2 \pi^{2}}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
    So, the result is: $- \frac{u^{2}}{4 \pi^{2}}$
  Now substitute $u$ back in:
  
  $- \frac{\operatorname{acot}^{2}{\left(2 x \right)}}{4 \pi^{2}}$
Method #2
1. Rewrite the integrand:
  
  $\frac{\operatorname{acot}{\left(2 x \right)}}{\pi^{2} \left(4 x^{2} + 1\right)} = \frac{\operatorname{acot}{\left(2 x \right)}}{4 \pi^{2} x^{2} + \pi^{2}}$
2. Let $u = \operatorname{acot}{\left(2 x \right)}$ .
  
  Then let $du = - \frac{2 dx}{4 x^{2} + 1}$ and substitute $- \frac{du}{2 \pi^{2}}$ :
  
  $\int \left(- \frac{u}{2 \pi^{2}}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int u\, du = - \frac{\int u\, du}{2 \pi^{2}}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
    So, the result is: $- \frac{u^{2}}{4 \pi^{2}}$
  Now substitute $u$ back in:
  
  $- \frac{\operatorname{acot}^{2}{\left(2 x \right)}}{4 \pi^{2}}$
Method #3
1. Rewrite the integrand:
  
  $\frac{\operatorname{acot}{\left(2 x \right)}}{\pi^{2} \left(4 x^{2} + 1\right)} = \frac{\operatorname{acot}{\left(2 x \right)}}{4 \pi^{2} x^{2} + \pi^{2}}$
2. Let $u = \operatorname{acot}{\left(2 x \right)}$ .
  
  Then let $du = - \frac{2 dx}{4 x^{2} + 1}$ and substitute $- \frac{du}{2 \pi^{2}}$ :
  
  $\int \left(- \frac{u}{2 \pi^{2}}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int u\, du = - \frac{\int u\, du}{2 \pi^{2}}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
    So, the result is: $- \frac{u^{2}}{4 \pi^{2}}$
  Now substitute $u$ back in:
  
  $- \frac{\operatorname{acot}^{2}{\left(2 x \right)}}{4 \pi^{2}}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                  
 |                             2     
 |   acot(2*x)             acot (2*x)
 | -------------- dx = C - ----------
 |   2 /   2    \                2   
 | pi *\4*x  + 1/            4*pi    
 |                                   
/

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

Simplify

The answer [src]

1/64

$$\frac{1}{64}$$

1/64

$$\frac{1}{64}$$

1/64

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3