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Derivative of:
arctan(x/4)
Identical expressions
arctan(x/ four)
arc tangent of (x divide by 4)
arc tangent of (x divide by four)
arctanx/4
arctan(x divide by 4)
arctan(x/4)dx

Integral of arctan(x/4) dx

The solution

You have entered [src]

     ___          
 4*\/ 3           
    /             
   |              
   |        /x\   
   |    atan|-| dx
   |        \4/   
   |              
  /               
  0

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

Integral(atan(x/4), (x, 0, 4*sqrt(3)))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \frac{x}{4}$ .
  
  Then let $du = \frac{dx}{4}$ and substitute $4 du$ :
  
  $\int 4 \operatorname{atan}{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \operatorname{atan}{\left(u \right)}\, du = 4 \int \operatorname{atan}{\left(u \right)}\, du$
    1. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(u \right)} = \operatorname{atan}{\left(u \right)}$ and let $\operatorname{dv}{\left(u \right)} = 1$ .
      
      Then $\operatorname{du}{\left(u \right)} = \frac{1}{u^{2} + 1}$ .
      
      To find $v{\left(u \right)}$ :
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{u}{u^{2} + 1}\, du = \frac{\int \frac{2 u}{u^{2} + 1}\, du}{2}$
      
      Let $u = u^{2} + 1$ .
      
      Then let $du = 2 u du$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{1}{2 u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(u^{2} + 1 \right)}$
      
      So, the result is: $\frac{\log{\left(u^{2} + 1 \right)}}{2}$
    So, the result is: $4 u \operatorname{atan}{\left(u \right)} - 2 \log{\left(u^{2} + 1 \right)}$
  Now substitute $u$ back in:
  
  $x \operatorname{atan}{\left(\frac{x}{4} \right)} - 2 \log{\left(\frac{x^{2}}{16} + 1 \right)}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \operatorname{atan}{\left(\frac{x}{4} \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .
  
  Then $\operatorname{du}{\left(x \right)} = \frac{1}{4 \left(\frac{x^{2}}{16} + 1\right)}$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, dx = x$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x}{4 \left(\frac{x^{2}}{16} + 1\right)}\, dx = \frac{\int \frac{x}{\frac{x^{2}}{16} + 1}\, dx}{4}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{x}{\frac{x^{2}}{16} + 1}\, dx = 8 \int \frac{x}{8 \left(\frac{x^{2}}{16} + 1\right)}\, dx$
    1. Let $u = \frac{x^{2}}{16} + 1$ .
      
      Then let $du = \frac{x dx}{8}$ and substitute $8 du$ :
      
      $\int \frac{8}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(\frac{x^{2}}{16} + 1 \right)}$
    So, the result is: $8 \log{\left(\frac{x^{2}}{16} + 1 \right)}$
  So, the result is: $2 \log{\left(\frac{x^{2}}{16} + 1 \right)}$
Now simplify:

$x \operatorname{atan}{\left(\frac{x}{4} \right)} - 2 \log{\left(\frac{x^{2}}{16} + 1 \right)}$
Add the constant of integration:

$x \operatorname{atan}{\left(\frac{x}{4} \right)} - 2 \log{\left(\frac{x^{2}}{16} + 1 \right)}+ \mathrm{constant}$

The answer is:

$x \operatorname{atan}{\left(\frac{x}{4} \right)} - 2 \log{\left(\frac{x^{2}}{16} + 1 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                          
 |                       /     2\            
 |     /x\               |    x |         /x\
 | atan|-| dx = C - 2*log|1 + --| + x*atan|-|
 |     \4/               \    16/         \4/
 |                                           
/

$$\int \operatorname{atan}{\left(\frac{x}{4} \right)}\, dx = C + x \operatorname{atan}{\left(\frac{x}{4} \right)} - 2 \log{\left(\frac{x^{2}}{16} + 1 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                                ___
                         4*pi*\/ 3 
-2*log(64) + 2*log(16) + ----------
                             3

$$- 2 \log{\left(64 \right)} + 2 \log{\left(16 \right)} + \frac{4 \sqrt{3} \pi}{3}$$

                                ___
                         4*pi*\/ 3 
-2*log(64) + 2*log(16) + ----------
                             3

$$- 2 \log{\left(64 \right)} + 2 \log{\left(16 \right)} + \frac{4 \sqrt{3} \pi}{3}$$

-2*log(64) + 2*log(16) + 4*pi*sqrt(3)/3

Numerical answer [src]

4.48260873469709

4.48260873469709

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

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How to use it?

Identical expressions

Integral of arctan(x/4) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2